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[Axiom-developer] 20080423.01.tpd.patch (CATS Schaums-Axiom equivalence
|
From: |
daly |
|
Subject: |
[Axiom-developer] 20080423.01.tpd.patch (CATS Schaums-Axiom equivalence testing (17-22)) |
|
Date: |
Thu, 24 Apr 2008 01:01:15 -0500 |
schaum17.input.pamphlet
14:339 Schaums and Axiom agree
14:340 Schaums and Axiom agree
14:341 Schaums and Axiom agree
14:342 Schaums and Axiom agree
14:343 Schaums and Axiom agree by definition
14:344 Axiom cannot compute this integral
14:345 Schaums and Axiom agree
14:346 Axiom cannot compute this integral
14:347 Schaums and Axiom agreee
14:348 Schaums and Axiom differ by a constant
14:349 Schaums and Axiom agree
14:350 Schaums and Axiom agree
14:351 Schaums and Axiom agree
14:352 Schaums and Axiom agree
14:353 Axiom cannot simplify this expression
14:354 Schaums and Axiom differ by a constant
14:355 Axiom cannot simplify this expression
14:356 Axiom cannot simplify this expression
14:357 Axiom cannot simplify this expression
14:358 Axiom cannot simplify this expression
14:359 Axiom cannot simplify this expression
14:360 Axiom cannot simplify these equations
14:361 Schaums and Axiom differ by a constant
14:362 Axiom cannot simplify this expression
14:363 Axiom cannot simplify this expression
14:364 Axiom cannot compute this integral
14:365 Axiom cannot compute this integral
14:366 Axiom cannot compute this integral
14:367 Axiom cannot compute this integral
14:368 Axiom cannot compute this integral
schaum18.input.pamphlet
14:369 Schaums and Axiom agree
14:370 Schaums and Axiom agree
14:371 Schaums and Axiom agree
14:372 Schaums and Axiom agree
14:373 Schaums and Axiom agree by definition
14:374 Axiom cannot compute this integral
14:375 Axiom cannot simplify these expressions
14:376 Axiom cannot compute this integral
14:377 Schaums and Axiom agree
14:378 Schaums and Axiom differ by a constant
14:379 Schaums and Axiom agree
14:380 Axiom cannot simplify this expression
14:381 Schaums and Axiom agree
14:382 Axiom cannot simplify this expression
14:383 Axiom cannot simplify this expression
14:384 Axiom cannot simplify this expression
14:385 Axiom cannot simplify this expression
14:386 Axiom cannot simplify this expression
14:387 Axiom cannot simplify this expression
14:388 Axiom cannot simplify this expression
14:389 Axiom cannot simplify this expression
14:390 Axiom cannot simplify these expressions
14:391 Schaums and Axiom agree
14:392 Axiom cannot simplify these expressions
14:393 Axiom cannot simplify these expressions
14:394 Axiom cannot compute this integral
14:395 Axiom cannot compute this integral
14:396 Axiom cannot compute this integral
14:397 Axiom cannot compute this integral
14:398 Axiom cannot compute this integral
schaum19.input.pamphlet
14:399 Schaums and Axiom differ by a constant
14:400 Axiom is unable to simplify this expression
14:401 Schaums and Axiom agree
14:402 Schaums and Axiom agree
14:403 Schaums and Axiom agree
14:404 Schaums and Axiom differ by a constant
14:405 Schaums and Axiom differ by a constant
14:406 Schaums and Axiom differ by a constant
14:407 Schaums and Axiom agree
14:408 Schaums and Axiom differ by a constant
14:409 Schaums and Axiom agree
14:410 Schaums and Axiom differ by a constant
14:411 Schaums and Axiom differ by a constant
14:412 Axiom cannot simplify this expression
14:413 Schaums and Axiom agree
14:414 Schaums and Axiom agree
14:415 Schaums and Axiom differ by a constant
14:416 Schaums and Axiom differ by a constant
14:417 Schaums and Axiom agree
14:418 Schaums and Axiom agree
14:419 Axiom cannot simplify this expression
14:420 Axiom cannot simplify these expressions
14:421 Schaums and Axiom agree
14:422 Axiom cannot simplify this expression
14:423 Axiom cannot simplify this expressions
14:424 Schaums and Axiom differ by a constant
14:425 Axiom cannot compute this integral
14:426 Axiom cannot compute this integral
14:427 Axiom cannot compute this integral
14:428 Axiom cannot compute this integral
schaum20.input.pamphlet
14:429 Schaums and Axiom agree
14:430 Schaums and Axiom agree
14:431 Schaums and Axiom agree
14:432 Schaums and Axiom agree
14:433 Schaums and Axiom differ by a constant
14:435 Axiom cannot compute this integral
14:436 Axiom cannot compute this integral
14:437 Schaums and Axiom agree
14:438 Schaums and Axiom agree
14:439 Axiom cannot compute this integral
schaum21.input.pamphlet
14:440 Schaums and Axiom agree
14:441 Schaums and Axiom agree
14:442 Schaums and Axiom agree
14:443 Schaums and Axiom agree
14:444 Schaums and Axiom differ by a constant
14:445 Schaums and Axiom agree
14:446 Axiom cannot compute this integral
14:447 Axiom cannot compute this integral
14:448 Schaums and Axiom agree
14:449 Axiom cannot simplify this expression
14:450 Axiom cannot compute this integral
schaum22.input.pamphlet
14:451 Schaums and Axiom differ by a constant
14:452 Schaums and Axiom agree
14:453 Schaums and Axiom differ by a constant
14:454 Axiom cannot simplify this expression
14:455 Schaums and Axiom agree
14:456 Axiom cannot compute this integral
14:457 Axiom cannot compute this integral
14:458 Schaums and Axiom differ by a constant
14:459 Schaums and Axiom agree
14:460 Axiom cannot compute this integral
========================================================================
diff --git a/changelog b/changelog
index 4b454b8..80e94f3 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,9 @@
+20080423 tpd src/input/schaum22.input show Schaums-Axiom equivalence
+20080423 tpd src/input/schaum21.input show Schaums-Axiom equivalence
+20080423 tpd src/input/schaum20.input show Schaums-Axiom equivalence
+20080423 tpd src/input/schaum19.input show Schaums-Axiom equivalence
+20080423 tpd src/input/schaum18.input show Schaums-Axiom equivalence
+20080423 tpd src/input/schaum17.input show Schaums-Axiom equivalence
20080421 tpd src/input/schaum16.input show Schaums-Axiom equivalence
20080421 tpd src/input/schaum15.input show Schaums-Axiom equivalence
20080421 tpd src/input/schaum14.input show Schaums-Axiom equivalence
diff --git a/src/input/schaum17.input.pamphlet
b/src/input/schaum17.input.pamphlet
index fd40132..a11b623 100644
--- a/src/input/schaum17.input.pamphlet
+++ b/src/input/schaum17.input.pamphlet
@@ -18,7 +18,7 @@ $$
)set message auto off
)clear all
---S 1 of 30
+--S 1
aa:=integrate(sin(a*x),x)
--R
--R
@@ -27,6 +27,22 @@ aa:=integrate(sin(a*x),x)
--R a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 2
+bb:=-cos(a*x)/a
+--R
+--R cos(a x)
+--R (2) - --------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 3 14:339 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.340~~~~~$\displaystyle
@@ -37,7 +53,7 @@ $$
<<*>>=
)clear all
---S 2 of 30
+--S 4
aa:=integrate(x*sin(a*x),x)
--R
--R
@@ -47,6 +63,23 @@ aa:=integrate(x*sin(a*x),x)
--R a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 5
+bb:=sin(a*x)/a^2-(x*cos(a*x))/a
+--R
+--R sin(a x) - a x cos(a x)
+--R (2) -----------------------
+--R 2
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 6 14:340 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.341~~~~~$\displaystyle
@@ -57,7 +90,7 @@ $$
<<*>>=
)clear all
---S 3 of 30
+--S 7
aa:=integrate(x^2*sin(a*x),x)
--R
--R
@@ -68,6 +101,24 @@ aa:=integrate(x^2*sin(a*x),x)
--R a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 8
+bb:=(2*x)/a^2*sin(a*x)+(2/a^3-x^2/a)*cos(a*x)
+--R
+--R 2 2
+--R 2a x sin(a x) + (- a x + 2)cos(a x)
+--R (2) ------------------------------------
+--R 3
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 9 14:341 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.342~~~~~$\displaystyle
@@ -79,7 +130,7 @@ $$
<<*>>=
)clear all
---S 4 of 30
+--S 10
aa:=integrate(x^3*sin(a*x),x)
--R
--R
@@ -90,6 +141,24 @@ aa:=integrate(x^3*sin(a*x),x)
--R a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 11
+bb:=((3*x^2)/a^2-6/a^4)*sin(a*x)+(6*x/a^3-x^3/a)*cos(a*x)
+--R
+--R 2 2 3 3
+--R (3a x - 6)sin(a x) + (- a x + 6a x)cos(a x)
+--R (2) ---------------------------------------------
+--R 4
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 12 14:342 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.343~~~~~$\displaystyle
@@ -100,7 +169,7 @@ $$
<<*>>=
)clear all
---S 5 of 30
+--S 13 14:343 Schaums and Axiom agree by definition
aa:=integrate(sin(x)/x,x)
--R
--R
@@ -117,7 +186,7 @@ $$
<<*>>=
)clear all
---S 6 of 30
+--S 14 14:344 Axiom cannot compute this integral
aa:=integrate(sin(a*x)/x^2,x)
--R
--R
@@ -139,7 +208,7 @@ $$
<<*>>=
)clear all
---S 7 of 30
+--S 15
aa:=integrate(1/sin(a*x),x)
--R
--R
@@ -150,6 +219,35 @@ aa:=integrate(1/sin(a*x),x)
--R a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 16
+bb:=1/a*log(tan((a*x)/2))
+--R
+--R a x
+--R log(tan(---))
+--R 2
+--R (2) -------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 17
+cc:=aa-bb
+--R
+--R a x sin(a x)
+--R - log(tan(---)) + log(------------)
+--R 2 cos(a x) + 1
+--R (3) -----------------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 18 14:345 Schaums and Axiom agree
+dd:=complexNormalize cc
+--R
+--R (4) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.346~~~~~$\displaystyle
@@ -162,7 +260,7 @@ $$
<<*>>=
)clear all
---S 8 of 30
+--S 19 14:346 Axiom cannot compute this integral
aa:=integrate(x/sin(a*x),x)
--R
--R
@@ -182,7 +280,7 @@ $$
<<*>>=
)clear all
---S 9 of 30
+--S 20
aa:=integrate(sin(a*x)^2,x)
--R
--R
@@ -191,6 +289,31 @@ aa:=integrate(sin(a*x)^2,x)
--R 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 21
+bb:=x/2-sin(2*a*x)/(4*a)
+--R
+--R - sin(2a x) + 2a x
+--R (2) ------------------
+--R 4a
+--R Type: Expression
Integer
+--E
+
+--S 22
+cc:=aa-bb
+--R
+--R sin(2a x) - 2cos(a x)sin(a x)
+--R (3) -----------------------------
+--R 4a
+--R Type: Expression
Integer
+--E
+
+--S 23 14:347 Schaums and Axiom agreee
+dd:=complexNormalize cc
+--R
+--R (4) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.348~~~~~$\displaystyle
@@ -201,7 +324,7 @@ $$
<<*>>=
)clear all
---S 10 of 30
+--S 24
aa:=integrate(x*sin(a*x)^2,x)
--R
--R
@@ -212,6 +335,38 @@ aa:=integrate(x*sin(a*x)^2,x)
--R 4a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 25
+bb:=x^2/4-(x*sin(2*a*x))/(4*a)-cos(2*a*x)/(8*a^2)
+--R
+--R 2 2
+--R - 2a x sin(2a x) - cos(2a x) + 2a x
+--R (2) ------------------------------------
+--R 2
+--R 8a
+--R Type: Expression
Integer
+--E
+
+--S 26
+cc:=aa-bb
+--R
+--R 2
+--R 2a x sin(2a x) - 4a x cos(a x)sin(a x) + cos(2a x) - 2cos(a x)
+--R (3) ---------------------------------------------------------------
+--R 2
+--R 8a
+--R Type: Expression
Integer
+--E
+
+--S 27 14:348 Schaums and Axiom differ by a constant
+dd:=complexNormalize cc
+--R
+--R 1
+--R (4) - ---
+--R 2
+--R 8a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.349~~~~~$\displaystyle
@@ -222,7 +377,7 @@ $$
<<*>>=
)clear all
---S 11 of 30
+--S 28
aa:=integrate(sin(a*x)^3,x)
--R
--R
@@ -232,6 +387,23 @@ aa:=integrate(sin(a*x)^3,x)
--R 3a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 29
+bb:=-cos(a*x)/a+cos(a*x)^3/(3*a)
+--R
+--R 3
+--R cos(a x) - 3cos(a x)
+--R (2) ---------------------
+--R 3a
+--R Type: Expression
Integer
+--E
+
+--S 30 14:349 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.350~~~~~$\displaystyle
@@ -242,7 +414,7 @@ $$
<<*>>=
)clear all
---S 12 of 30
+--S 31
aa:=integrate(sin(a*x)^4,x)
--R
--R
@@ -252,6 +424,32 @@ aa:=integrate(sin(a*x)^4,x)
--R 8a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 32
+bb:=(3*x)/8-sin(2*a*x)/(4*a)+sin(4*a*x)/(32*a)
+--R
+--R sin(4a x) - 8sin(2a x) + 12a x
+--R (2) ------------------------------
+--R 32a
+--R Type: Expression
Integer
+--E
+
+--S 33
+cc:=aa-bb
+--R
+--R 3
+--R - sin(4a x) + 8sin(2a x) + (8cos(a x) - 20cos(a x))sin(a x)
+--R (3) ------------------------------------------------------------
+--R 32a
+--R Type: Expression
Integer
+--E
+
+--S 34 14:350 Schaums and Axiom agree
+dd:=complexNormalize cc
+--R
+--R (4) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.351~~~~~$\displaystyle
@@ -262,7 +460,7 @@ $$
<<*>>=
)clear all
---S 13 of 30
+--S 35
aa:=integrate(1/sin(a*x)^2,x)
--R
--R
@@ -271,6 +469,31 @@ aa:=integrate(1/sin(a*x)^2,x)
--R a sin(a x)
--R Type: Union(Expression
Integer,...)
--E
+
+--S 36
+bb:=-1/a*cot(a*x)
+--R
+--R cot(a x)
+--R (2) - --------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 37
+cc:=aa-bb
+--R
+--R cot(a x)sin(a x) - cos(a x)
+--R (3) ---------------------------
+--R a sin(a x)
+--R Type: Expression
Integer
+--E
+
+--S 38 14:351 Schaums and Axiom agree
+dd:=complexNormalize cc
+--R
+--R (4) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.352~~~~~$\displaystyle
@@ -281,7 +504,7 @@ $$
<<*>>=
)clear all
---S 14 of 30
+--S 39
aa:=integrate(1/sin(a*x)^3,x)
--R
--R
@@ -293,6 +516,66 @@ aa:=integrate(1/sin(a*x)^3,x)
--R 2a cos(a x) - 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 40
+bb:=-cos(a*x)/(2*a*sin(a*x)^2)+1/(2*a)*log(tan((a*x)/2))
+--R
+--R 2 a x
+--R sin(a x) log(tan(---)) - cos(a x)
+--R 2
+--R (2) ---------------------------------
+--R 2
+--R 2a sin(a x)
+--R Type: Expression
Integer
+--E
+
+--S 41
+cc:=aa-bb
+--R
+--R (3)
+--R 2 2 a x
+--R (- cos(a x) + 1)sin(a x) log(tan(---))
+--R 2
+--R +
+--R 2 2 sin(a x) 2
3
+--R (cos(a x) - 1)sin(a x) log(------------) + cos(a x)sin(a x) +
cos(a x)
+--R cos(a x) + 1
+--R +
+--R - cos(a x)
+--R /
+--R 2 2
+--R (2a cos(a x) - 2a)sin(a x)
+--R Type: Expression
Integer
+--E
+
+--S 42
+dd:=expandLog cc
+--R
+--R (4)
+--R 2 2 a x
+--R (- cos(a x) + 1)sin(a x) log(tan(---))
+--R 2
+--R +
+--R 2 2
+--R (cos(a x) - 1)sin(a x) log(sin(a x))
+--R +
+--R 2 2 2
+--R (- cos(a x) + 1)sin(a x) log(cos(a x) + 1) + cos(a x)sin(a x)
+--R +
+--R 3
+--R cos(a x) - cos(a x)
+--R /
+--R 2 2
+--R (2a cos(a x) - 2a)sin(a x)
+--R Type: Expression
Integer
+--E
+
+--S 43 14:352 Schaums and Axiom agree
+ee:=complexNormalize dd
+--R
+--R (5) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.353~~~~~$\displaystyle
@@ -303,7 +586,7 @@ $$
<<*>>=
)clear all
---S 15 of 30
+--S 44
aa:=integrate(sin(p*x)*sin(q*x),x)
--R
--R
@@ -312,7 +595,30 @@ aa:=integrate(sin(p*x)*sin(q*x),x)
--R 2 2
--R q - p
--R Type: Union(Expression
Integer,...)
+--E
+
+--S 45
+bb:=(sin(p-q)*x)/(2*(p-q))-(sin(p+q)*x)/(2*(p+q))
+--R
+--R (- q + p)x sin(q + p) + (q + p)x sin(q - p)
+--R (2) -------------------------------------------
+--R 2 2
+--R 2q - 2p
+--R Type: Expression
Integer
--E
+
+--S 46 14:353 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R 2p cos(p x)sin(q x) - 2q cos(q x)sin(p x) + (q - p)x sin(q + p)
+--R +
+--R (- q - p)x sin(q - p)
+--R /
+--R 2 2
+--R 2q - 2p
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.354~~~~~$\displaystyle
@@ -323,7 +629,7 @@ $$
<<*>>=
)clear all
---S 16 of 30
+--S 47
aa:=integrate(1/(1-sin(a*x)),x)
--R
--R
@@ -331,7 +637,38 @@ aa:=integrate(1/(1-sin(a*x)),x)
--R (1) ---------------------------
--R a sin(a x) - a cos(a x) - a
--R Type: Union(Expression
Integer,...)
+--E
+
+--S 48
+bb:=1/a*tan(%pi/4+(a*x)/2)
+--R
+--R 2a x + %pi
+--R tan(----------)
+--R 4
+--R (2) ---------------
+--R a
+--R Type: Expression
Integer
--E
+
+--S 49
+cc:=aa-bb
+--R
+--R 2a x + %pi
+--R (- sin(a x) + cos(a x) + 1)tan(----------) - 2cos(a x) - 2
+--R 4
+--R (3) ----------------------------------------------------------
+--R a sin(a x) - a cos(a x) - a
+--R Type: Expression
Integer
+--E
+
+--S 50 14:354 Schaums and Axiom differ by a constant
+dd:=complexNormalize cc
+--R
+--R 1
+--R (4) -
+--R a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.355~~~~~$\displaystyle
@@ -343,7 +680,7 @@ $$
<<*>>=
)clear all
---S 17 of 30
+--S 51
aa:=integrate(x/(1-sin(ax)),x)
--R
--R
@@ -353,6 +690,35 @@ aa:=integrate(x/(1-sin(ax)),x)
--R 2sin(ax) - 2
--R Type: Union(Expression
Integer,...)
--E
+
+--S 52
+bb:=x/a*tan(%pi/4+(a*x)/2)+2/a^2*log(sin(%pi/4-(a*x)/2))
+--R
+--R 2a x - %pi 2a x + %pi
+--R 2log(- sin(----------)) + a x tan(----------)
+--R 4 4
+--R (2) ---------------------------------------------
+--R 2
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 53 14:355 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R 2a x - %pi
+--R (- 4sin(ax) + 4)log(- sin(----------))
+--R 4
+--R +
+--R 2a x + %pi 2 2
+--R (- 2a x sin(ax) + 2a x)tan(----------) - a x
+--R 4
+--R /
+--R 2 2
+--R 2a sin(ax) - 2a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.356~~~~~$\displaystyle
@@ -363,7 +729,7 @@ $$
<<*>>=
)clear all
---S 18 of 30
+--S 54
aa:=integrate(1/(1+sin(ax)),x)
--R
--R
@@ -372,6 +738,46 @@ aa:=integrate(1/(1+sin(ax)),x)
--R sin(ax) + 1
--R Type: Union(Expression
Integer,...)
--E
+
+--S 55
+bb:=-1/a*tan(%pi/4-(a*x)/2)
+--R
+--R 2a x - %pi
+--R tan(----------)
+--R 4
+--R (2) ---------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 56
+cc:=aa-bb
+--R
+--R 2a x - %pi
+--R (- sin(ax) - 1)tan(----------) + a x
+--R 4
+--R (3) ------------------------------------
+--R a sin(ax) + a
+--R Type: Expression
Integer
+--E
+
+--S 57
+tanrule:=rule(tan(a/b) == sin(a)/cos(b))
+--R
+--R a sin(a)
+--R (4) tan(-) == ------
+--R b cos(b)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 58 14:356 Axiom cannot simplify this expression
+dd:=tanrule cc
+--R
+--R (- sin(ax) - 1)sin(2a x - %pi) + a x cos(4)
+--R (5) -------------------------------------------
+--R a cos(4)sin(ax) + a cos(4)
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.357~~~~~$\displaystyle
@@ -383,7 +789,7 @@ $$
<<*>>=
)clear all
---S 19 of 30
+--S 59
aa:=integrate(x/(1+sin(a*x)),x)
--R
--R
@@ -402,6 +808,100 @@ aa:=integrate(x/(1+sin(a*x)),x)
--R a sin(a x) + a cos(a x) + a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 60
+bb:=-x/a*tan(%pi/4-(a*x)/2)+2/a^2*log(sin(%pi/4+(a*x)/2))
+--R
+--R 2a x + %pi 2a x - %pi
+--R 2log(sin(----------)) + a x tan(----------)
+--R 4 4
+--R (2) -------------------------------------------
+--R 2
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 61
+cc:=aa-bb
+--R
+--R (3)
+--R sin(a x) + cos(a x) + 1
+--R (2sin(a x) + 2cos(a x) + 2)log(-----------------------)
+--R cos(a x) + 1
+--R +
+--R 2a x + %pi
+--R (- 2sin(a x) - 2cos(a x) - 2)log(sin(----------))
+--R 4
+--R +
+--R 2
+--R (- sin(a x) - cos(a x) - 1)log(------------)
+--R cos(a x) + 1
+--R +
+--R 2a x - %pi
+--R (- a x sin(a x) - a x cos(a x) - a x)tan(----------) + a x sin(a x)
+--R 4
+--R +
+--R - a x cos(a x) - a x
+--R /
+--R 2 2 2
+--R a sin(a x) + a cos(a x) + a
+--R Type: Expression
Integer
+--E
+
+--S 62
+dd:=expandLog cc
+--R
+--R (4)
+--R (2sin(a x) + 2cos(a x) + 2)log(sin(a x) + cos(a x) + 1)
+--R +
+--R 2a x + %pi
+--R (- 2sin(a x) - 2cos(a x) - 2)log(sin(----------))
+--R 4
+--R +
+--R (- sin(a x) - cos(a x) - 1)log(cos(a x) + 1)
+--R +
+--R 2a x - %pi
+--R (- a x sin(a x) - a x cos(a x) - a x)tan(----------)
+--R 4
+--R +
+--R (- log(2) + a x)sin(a x) + (- log(2) - a x)cos(a x) - log(2) - a x
+--R /
+--R 2 2 2
+--R a sin(a x) + a cos(a x) + a
+--R Type: Expression
Integer
+--E
+
+--S 63
+tanrule:=rule(tan(a/b) == sin(a)/cos(b))
+--R
+--R a sin(a)
+--R (5) tan(-) == ------
+--R b cos(b)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 64 14:357 Axiom cannot simplify this expression
+ee:=tanrule dd
+--R
+--R (6)
+--R (2cos(4)sin(a x) + 2cos(4)cos(a x) + 2cos(4))log(sin(a x) + cos(a x)
+ 1)
+--R +
+--R 2a x + %pi
+--R (- 2cos(4)sin(a x) - 2cos(4)cos(a x) - 2cos(4))log(sin(----------))
+--R 4
+--R +
+--R (- cos(4)sin(a x) - cos(4)cos(a x) - cos(4))log(cos(a x) + 1)
+--R +
+--R (- a x sin(a x) - a x cos(a x) - a x)sin(2a x - %pi)
+--R +
+--R (- cos(4)log(2) + a x cos(4))sin(a x)
+--R +
+--R (- cos(4)log(2) - a x cos(4))cos(a x) - cos(4)log(2) - a x cos(4)
+--R /
+--R 2 2 2
+--R a cos(4)sin(a x) + a cos(4)cos(a x) + a cos(4)
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.358~~~~~$\displaystyle
@@ -413,7 +913,7 @@ $$
<<*>>=
)clear all
---S 20 of 30
+--S 65
aa:=integrate(1/(1-sin(a*x))^2,x)
--R
--R
@@ -423,7 +923,76 @@ aa:=integrate(1/(1-sin(a*x))^2,x)
--R 2
--R (3a cos(a x) + 6a)sin(a x) + 3a cos(a x) - 3a cos(a x) - 6a
--R Type: Union(Expression
Integer,...)
+--E
+
+--S 66
+bb:=-1/(2*a)*tan(%pi/4+(a*x)/2)+1/(6*a)*tan(%pi/4+(a*x)/2)^3
+--R
+--R 2a x + %pi 3 2a x + %pi
+--R tan(----------) - 3tan(----------)
+--R 4 4
+--R (2) -----------------------------------
+--R 6a
+--R Type: Expression
Integer
--E
+
+--S 67
+cc:=aa-bb
+--R
+--R (3)
+--R 2 2a x + %pi 3
+--R ((- cos(a x) - 2)sin(a x) - cos(a x) + cos(a x) + 2)tan(----------)
+--R 4
+--R +
+--R 2 2a x + %pi
+--R ((3cos(a x) + 6)sin(a x) + 3cos(a x) - 3cos(a x) - 6)tan(----------)
+--R 4
+--R +
+--R 2
+--R (6cos(a x) + 6)sin(a x) + 2cos(a x) - 8cos(a x) - 10
+--R /
+--R 2
+--R (6a cos(a x) + 12a)sin(a x) + 6a cos(a x) - 6a cos(a x) - 12a
+--R Type: Expression
Integer
+--E
+
+--S 68
+tanrule:=rule(tan(a/b) == sin(a)/cos(b))
+--R
+--R a sin(a)
+--R (4) tan(-) == ------
+--R b cos(b)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 69 14:358 Axiom cannot simplify this expression
+dd:=tanrule cc
+--R
+--R (5)
+--R 2 3
+--R ((- cos(a x) - 2)sin(a x) - cos(a x) + cos(a x) + 2)sin(2a x + %pi)
+--R +
+--R 2 2 2 2
+--R (3cos(4) cos(a x) + 6cos(4) )sin(a x) + 3cos(4) cos(a x)
+--R +
+--R 2 2
+--R - 3cos(4) cos(a x) - 6cos(4)
+--R *
+--R sin(2a x + %pi)
+--R +
+--R 3 3 3 2
+--R (6cos(4) cos(a x) + 6cos(4) )sin(a x) + 2cos(4) cos(a x)
+--R +
+--R 3 3
+--R - 8cos(4) cos(a x) - 10cos(4)
+--R /
+--R 3 3 3 2
+--R (6a cos(4) cos(a x) + 12a cos(4) )sin(a x) + 6a cos(4) cos(a x)
+--R +
+--R 3 3
+--R - 6a cos(4) cos(a x) - 12a cos(4)
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.359~~~~~$\displaystyle
@@ -435,7 +1004,7 @@ $$
<<*>>=
)clear all
---S 21 of 30
+--S 70
aa:=integrate(1/(1+sin(a*x))^2,x)
--R
--R
@@ -445,7 +1014,38 @@ aa:=integrate(1/(1+sin(a*x))^2,x)
--R 2
--R (3a cos(a x) + 6a)sin(a x) - 3a cos(a x) + 3a cos(a x) + 6a
--R Type: Union(Expression
Integer,...)
+--E
+
+--S 71
+bb:=-1/(2*a)*tan(%pi/4-(a*x)/2)-1/(6*a)*tan(%pi/4-(a*x)/2)^3
+--R
+--R 2a x - %pi 3 2a x - %pi
+--R tan(----------) + 3tan(----------)
+--R 4 4
+--R (2) -----------------------------------
+--R 6a
+--R Type: Expression
Integer
--E
+
+--S 72 14:359 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R 2 2a x - %pi 3
+--R ((- cos(a x) - 2)sin(a x) + cos(a x) - cos(a x) - 2)tan(----------)
+--R 4
+--R +
+--R 2 2a x -
%pi
+--R ((- 3cos(a x) - 6)sin(a x) + 3cos(a x) - 3cos(a x) -
6)tan(----------)
+--R 4
+--R +
+--R 2
+--R (- 6cos(a x) - 6)sin(a x) + 2cos(a x) - 8cos(a x) - 10
+--R /
+--R 2
+--R (6a cos(a x) + 12a)sin(a x) - 6a cos(a x) + 6a cos(a x) + 12a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.360~~~~~$\displaystyle
@@ -466,7 +1066,7 @@ $$
<<*>>=
)clear all
---S 22 of 30
+--S 73
aa:=integrate(1/(p+q*sin(a*x)),x)
--R
--R
@@ -498,6 +1098,156 @@ aa:=integrate(1/(p+q*sin(a*x)),x)
--R a\|- q + p
--R Type: Union(List Expression
Integer,...)
--E
+
+--S 74
+bb1:=2/(a*sqrt(p^2-q^2))*atan((p*tan((a*x)/2+q))/sqrt(p^2-q^2))
+--R
+--R a x + 2q
+--R p tan(--------)
+--R 2
+--R 2atan(---------------)
+--R +---------+
+--R | 2 2
+--R \|- q + p
+--R (2) ----------------------
+--R +---------+
+--R | 2 2
+--R a\|- q + p
+--R Type: Expression
Integer
+--E
+
+--S 75
+bb2:=1/(a*sqrt(q^2-p^2))*log((p*tan((a*x)/2+q-sqrt(q^2-p^2)))/(p*tan((a*x)/2+q+sqrt(q^2-p^2))))
+--R
+--R +-------+
+--R | 2 2
+--R 2\|q - p - a x - 2q
+--R tan(----------------------)
+--R 2
+--R log(- ---------------------------)
+--R +-------+
+--R | 2 2
+--R 2\|q - p + a x + 2q
+--R tan(----------------------)
+--R 2
+--R (3) ----------------------------------
+--R +-------+
+--R | 2 2
+--R a\|q - p
+--R Type: Expression
Integer
+--E
+
+--S 76
+cc1:=aa.1-bb1
+--R
+--R (4)
+--R +---------+
+--R | 2 2
+--R \|- q + p
+--R *
+--R log
+--R +-------+
+--R 2 2 2 | 2 2
+--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p
+--R +
+--R 2 3 3 2 3 2
+--R (- p q + p )sin(a x) + (- q + p q)cos(a x) - q + p q
+--R /
+--R q sin(a x) + p
+--R +
+--R a x + 2q
+--R +-------+ p tan(--------)
+--R | 2 2 2
+--R - 2\|q - p atan(---------------)
+--R +---------+
+--R | 2 2
+--R \|- q + p
+--R /
+--R +---------+ +-------+
+--R | 2 2 | 2 2
+--R a\|- q + p \|q - p
+--R Type: Expression
Integer
+--E
+
+--S 77
+cc2:=aa.2-bb1
+--R
+--R (5)
+--R +---------+ a x + 2q
+--R | 2 2 p
tan(--------)
+--R (p sin(a x) + q cos(a x) + q)\|- q + p 2
+--R - 2atan(-----------------------------------------) -
2atan(---------------)
+--R 2 2 2 2 +---------+
+--R (q - p )cos(a x) + q - p | 2 2
+--R \|- q + p
+--R
---------------------------------------------------------------------------
+--R +---------+
+--R | 2 2
+--R a\|- q + p
+--R Type: Expression
Integer
+--E
+
+--S 78
+cc3:=aa.1-bb2
+--R
+--R (6)
+--R +-------+
+--R | 2 2
+--R 2\|q - p - a x - 2q
+--R tan(----------------------)
+--R 2
+--R - log(- ---------------------------)
+--R +-------+
+--R | 2 2
+--R 2\|q - p + a x + 2q
+--R tan(----------------------)
+--R 2
+--R +
+--R log
+--R +-------+
+--R 2 2 2 | 2 2
+--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p
+--R +
+--R 2 3 3 2 3 2
+--R (- p q + p )sin(a x) + (- q + p q)cos(a x) - q + p q
+--R /
+--R q sin(a x) + p
+--R /
+--R +-------+
+--R | 2 2
+--R a\|q - p
+--R Type: Expression
Integer
+--E
+
+--S 79 14:360 Axiom cannot simplify these equations
+cc4:=aa.2-bb2
+--R
+--R (7)
+--R +-------+
+--R | 2 2
+--R 2\|q - p - a x - 2q
+--R +---------+ tan(----------------------)
+--R | 2 2 2
+--R - \|- q + p log(- ---------------------------)
+--R +-------+
+--R | 2 2
+--R 2\|q - p + a x + 2q
+--R tan(----------------------)
+--R 2
+--R +
+--R +---------+
+--R +-------+ | 2 2
+--R | 2 2 (p sin(a x) + q cos(a x) + q)\|- q + p
+--R - 2\|q - p atan(-----------------------------------------)
+--R 2 2 2 2
+--R (q - p )cos(a x) + q - p
+--R /
+--R +---------+ +-------+
+--R | 2 2 | 2 2
+--R a\|- q + p \|q - p
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.361~~~~~$\displaystyle
@@ -509,7 +1259,7 @@ $$
<<*>>=
)clear all
---S 23 of 30
+--S 80
aa:=integrate(1/(p+q*sin(a*x))^2,x)
--R
--R
@@ -554,6 +1304,206 @@ aa:=integrate(1/(p+q*sin(a*x))^2,x)
--R ]
--R Type: Union(List Expression
Integer,...)
--E
+
+--S 81
+t1:=integrate(1/(p+q*sin(a*x)),x)
+--R
+--R (2)
+--R [
+--R log
+--R +-------+
+--R 2 2 2 | 2 2
+--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p
+--R +
+--R 2 3 3 2 3 2
+--R (- p q + p )sin(a x) + (- q + p q)cos(a x) - q + p q
+--R /
+--R q sin(a x) + p
+--R /
+--R +-------+
+--R | 2 2
+--R a\|q - p
+--R ,
+--R +---------+
+--R | 2 2
+--R (p sin(a x) + q cos(a x) + q)\|- q + p
+--R 2atan(-----------------------------------------)
+--R 2 2 2 2
+--R (q - p )cos(a x) + q - p
+--R - ------------------------------------------------]
+--R +---------+
+--R | 2 2
+--R a\|- q + p
+--R Type: Union(List Expression
Integer,...)
+--E
+
+--S 82
+bb1:=(q*cos(a*x))/(a*(p^2-q^2)*(p+q*sin(a*x)))+p/(p^2-q^2)*t1.1
+--R
+--R (3)
+--R 2
+--R (- p q sin(a x) - p )
+--R *
+--R log
+--R +-------+
+--R 2 2 2 | 2 2
+--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p
+--R +
+--R 2 3 3 2 3 2
+--R (- p q + p )sin(a x) + (- q + p q)cos(a x) - q + p q
+--R /
+--R q sin(a x) + p
+--R +
+--R +-------+
+--R | 2 2
+--R - q cos(a x)\|q - p
+--R /
+--R +-------+
+--R 3 2 2 3 | 2 2
+--R ((a q - a p q)sin(a x) + a p q - a p )\|q - p
+--R Type: Expression
Integer
+--E
+
+--S 83
+bb2:=(q*cos(a*x))/(a*(p^2-q^2)*(p+q*sin(a*x)))+p/(p^2-q^2)*t1.2
+--R
+--R (4)
+--R +---------+
+--R | 2 2
+--R 2 (p sin(a x) + q cos(a x) + q)\|- q + p
+--R (2p q sin(a x) + 2p )atan(-----------------------------------------)
+--R 2 2 2 2
+--R (q - p )cos(a x) + q - p
+--R +
+--R +---------+
+--R | 2 2
+--R - q cos(a x)\|- q + p
+--R /
+--R +---------+
+--R 3 2 2 3 | 2 2
+--R ((a q - a p q)sin(a x) + a p q - a p )\|- q + p
+--R Type: Expression
Integer
+--E
+
+--S 84
+cc1:=aa.1-bb1
+--R
+--R (5)
+--R 2
+--R p
+--R *
+--R log
+--R +-------+
+--R 2 2 2 | 2 2
+--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p
+--R +
+--R 2 3 3 2 3 2
+--R (p q - p )sin(a x) + (q - p q)cos(a x) + q - p q
+--R /
+--R q sin(a x) + p
+--R +
+--R 2
+--R p
+--R *
+--R log
+--R +-------+
+--R 2 2 2 | 2 2
+--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p
+--R +
+--R 2 3 3 2 3 2
+--R (- p q + p )sin(a x) + (- q + p q)cos(a x) - q + p q
+--R /
+--R q sin(a x) + p
+--R +
+--R +-------+
+--R | 2 2
+--R - q\|q - p
+--R /
+--R +-------+
+--R 2 3 | 2 2
+--R (a p q - a p )\|q - p
+--R Type: Expression
Integer
+--E
+
+--S 85
+cc2:=aa.2-bb1
+--R
+--R (6)
+--R +---------+
+--R 2 | 2 2
+--R p \|- q + p
+--R *
+--R log
+--R +-------+
+--R 2 2 2 | 2 2
+--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p
+--R +
+--R 2 3 3 2 3 2
+--R (- p q + p )sin(a x) + (- q + p q)cos(a x) - q + p q
+--R /
+--R q sin(a x) + p
+--R +
+--R +---------+
+--R +-------+ | 2 2
+--R 2 | 2 2 (p sin(a x) + q cos(a x) + q)\|- q + p
+--R 2p \|q - p atan(-----------------------------------------)
+--R 2 2 2 2
+--R (q - p )cos(a x) + q - p
+--R +
+--R +---------+ +-------+
+--R | 2 2 | 2 2
+--R - q\|- q + p \|q - p
+--R /
+--R +---------+ +-------+
+--R 2 3 | 2 2 | 2 2
+--R (a p q - a p )\|- q + p \|q - p
+--R Type: Expression
Integer
+--E
+
+--S 86
+cc3:=aa.1-bb2
+--R
+--R (7)
+--R +---------+
+--R 2 | 2 2
+--R p \|- q + p
+--R *
+--R log
+--R +-------+
+--R 2 2 2 | 2 2
+--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p
+--R +
+--R 2 3 3 2 3 2
+--R (p q - p )sin(a x) + (q - p q)cos(a x) + q - p q
+--R /
+--R q sin(a x) + p
+--R +
+--R +---------+
+--R +-------+ | 2 2
+--R 2 | 2 2 (p sin(a x) + q cos(a x) + q)\|- q + p
+--R - 2p \|q - p atan(-----------------------------------------)
+--R 2 2 2 2
+--R (q - p )cos(a x) + q - p
+--R +
+--R +---------+ +-------+
+--R | 2 2 | 2 2
+--R - q\|- q + p \|q - p
+--R /
+--R +---------+ +-------+
+--R 2 3 | 2 2 | 2 2
+--R (a p q - a p )\|- q + p \|q - p
+--R Type: Expression
Integer
+--E
+
+--S 87 14:361 Schaums and Axiom differ by a constant
+cc4:=aa.2-bb2
+--R
+--R q
+--R (8) - -------------
+--R 2 3
+--R a p q - a p
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.362~~~~~$\displaystyle
@@ -564,7 +1514,7 @@ $$
<<*>>=
)clear all
---S 24 of 30
+--S 88
aa:=integrate(1/(p^2+a^2*sin(a*x)),x)
--R
--R
@@ -597,6 +1547,76 @@ aa:=integrate(1/(p^2+a^2*sin(a*x)),x)
--R a\|p - a
--R Type: Union(List Expression
Integer,...)
--E
+
+--S 89
+bb:=1/(a*p*sqrt(p^2+q^2))*atan((sqrt(p^2+q^2)*tan(a*x))/p)
+--R
+--R +-------+
+--R | 2 2
+--R tan(a x)\|q + p
+--R atan(------------------)
+--R p
+--R (2) ------------------------
+--R +-------+
+--R | 2 2
+--R a p\|q + p
+--R Type: Expression
Integer
+--E
+
+--S 90
+cc1:=aa.1-bb
+--R
+--R (3)
+--R +-------+
+--R | 2 2
+--R p\|q + p
+--R *
+--R log
+--R +---------+
+--R 2 2 4 4 4 | 4 4
+--R (a p sin(a x) + (- p + a )cos(a x) + a )\|- p + a
+--R +
+--R 6 4 2 2 4 6 2 4 6
+--R (p - a p )sin(a x) + (a p - a )cos(a x) + a p - a
+--R /
+--R 2 2
+--R a sin(a x) + p
+--R +
+--R +-------+
+--R +---------+ | 2 2
+--R | 4 4 tan(a x)\|q + p
+--R - \|- p + a atan(------------------)
+--R p
+--R /
+--R +---------+ +-------+
+--R | 4 4 | 2 2
+--R a p\|- p + a \|q + p
+--R Type: Expression
Integer
+--E
+
+--S 91 14:362 Axiom cannot simplify this expression
+cc2:=aa.2-bb
+--R
+--R (4)
+--R +-------+
+--R +-------+ | 2 2
+--R | 4 4 tan(a x)\|q + p
+--R - \|p - a atan(------------------)
+--R p
+--R +
+--R +-------+
+--R +-------+ 2 2 2 | 4 4
+--R | 2 2 (p sin(a x) + a cos(a x) + a )\|p - a
+--R 2p\|q + p atan(----------------------------------------)
+--R 4 4 4 4
+--R (p - a )cos(a x) + p - a
+--R /
+--R +-------+ +-------+
+--R | 4 4 | 2 2
+--R a p\|p - a \|q + p
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.363~~~~~$\displaystyle
@@ -616,7 +1636,7 @@ $$
<<*>>=
)clear all
---S 25 of 30
+--S 92
aa:=integrate(1/(p^2-q^2*sin(a*x)^2),x)
--R
--R
@@ -658,6 +1678,154 @@ aa:=integrate(1/(p^2-q^2*sin(a*x)^2),x)
--R ]
--R Type: Union(List Expression
Integer,...)
--E
+
+--S 93
+bb1:=1/(a*p*sqrt(p^2-q^2))*atan((sqrt(p^2-q^2)*tan(a*x))/p)
+--R
+--R +---------+
+--R | 2 2
+--R tan(a x)\|- q + p
+--R atan(--------------------)
+--R p
+--R (2) --------------------------
+--R +---------+
+--R | 2 2
+--R a p\|- q + p
+--R Type: Expression
Integer
+--E
+
+--S 94
+bb2:=1/(2*a*p*sqrt(q^2-p^2))*log((sqrt(q^2-p^2)*tan(a*x)+p)/(sqrt(q^2-p^2)*tan(a*x)-p))
+--R
+--R +-------+
+--R | 2 2
+--R tan(a x)\|q - p + p
+--R log(----------------------)
+--R +-------+
+--R | 2 2
+--R tan(a x)\|q - p - p
+--R (3) ---------------------------
+--R +-------+
+--R | 2 2
+--R 2a p\|q - p
+--R Type: Expression
Integer
+--E
+
+--S 95
+cc1:=aa.1-bb1
+--R
+--R (4)
+--R +---------+
+--R | 2 2
+--R \|- q + p
+--R *
+--R log
+--R +-------+
+--R 2 2 2 2 2 | 2 2
+--R ((- q + 2p )cos(a x) + q - p )\|q - p
+--R +
+--R 2 3
+--R (2p q - 2p )cos(a x)sin(a x)
+--R /
+--R 2 2 2 2
+--R q cos(a x) - q + p
+--R +
+--R +---------+
+--R +-------+ | 2 2
+--R | 2 2 tan(a x)\|- q + p
+--R - 2\|q - p atan(--------------------)
+--R p
+--R /
+--R +---------+ +-------+
+--R | 2 2 | 2 2
+--R 2a p\|- q + p \|q - p
+--R Type: Expression
Integer
+--E
+
+--S 96
+cc2:=aa.2-bb1
+--R
+--R (5)
+--R +---------+ +---------+
+--R | 2 2 | 2 2
+--R tan(a x)\|- q + p p sin(a x)\|- q + p
+--R - atan(--------------------) - atan(-------------------------------)
+--R p 2 2 2 2
+--R (2q - 2p )cos(a x) + 2q - 2p
+--R +
+--R 2 2 2 2
+--R ((2q - p )cos(a x) + 2q - 2p )sin(a x)
+--R - atan(-------------------------------------------)
+--R +---------+
+--R 2 | 2 2
+--R (p cos(a x) + 2p cos(a x) + p)\|- q + p
+--R /
+--R +---------+
+--R | 2 2
+--R a p\|- q + p
+--R Type: Expression
Integer
+--E
+
+--S 97
+cc3:=aa.1-bb2
+--R
+--R (6)
+--R +-------+
+--R | 2 2
+--R tan(a x)\|q - p + p
+--R - log(----------------------)
+--R +-------+
+--R | 2 2
+--R tan(a x)\|q - p - p
+--R +
+--R log
+--R +-------+
+--R 2 2 2 2 2 | 2 2
+--R ((- q + 2p )cos(a x) + q - p )\|q - p
+--R +
+--R 2 3
+--R (2p q - 2p )cos(a x)sin(a x)
+--R /
+--R 2 2 2 2
+--R q cos(a x) - q + p
+--R /
+--R +-------+
+--R | 2 2
+--R 2a p\|q - p
+--R Type: Expression
Integer
+--E
+
+--S 98 14:363 Axiom cannot simplify this expression
+cc4:=aa.2-bb2
+--R
+--R (7)
+--R +-------+
+--R +---------+ | 2 2
+--R | 2 2 tan(a x)\|q - p + p
+--R - \|- q + p log(----------------------)
+--R +-------+
+--R | 2 2
+--R tan(a x)\|q - p - p
+--R +
+--R +---------+
+--R +-------+ | 2 2
+--R | 2 2 p sin(a x)\|- q + p
+--R - 2\|q - p atan(-------------------------------)
+--R 2 2 2 2
+--R (2q - 2p )cos(a x) + 2q - 2p
+--R +
+--R +-------+ 2 2 2 2
+--R | 2 2 ((2q - p )cos(a x) + 2q - 2p )sin(a x)
+--R - 2\|q - p atan(-------------------------------------------)
+--R +---------+
+--R 2 | 2 2
+--R (p cos(a x) + 2p cos(a x) + p)\|- q + p
+--R /
+--R +---------+ +-------+
+--R | 2 2 | 2 2
+--R 2a p\|- q + p \|q - p
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.364~~~~~$\displaystyle
@@ -669,7 +1837,7 @@ $$
<<*>>=
)clear all
---S 26 of 30
+--S 99 14:364 Axiom cannot compute this integral
aa:=integrate(x^m*sin(a*x),x)
--R
--R
@@ -689,7 +1857,7 @@ $$
<<*>>=
)clear all
---S 26 of 30
+--S 100 14:365 Axiom cannot compute this integral
aa:=integrate(sin(a*x)/x^n,x)
--R
--R
@@ -710,7 +1878,7 @@ $$
<<*>>=
)clear all
---S 28 of 30
+--S 101 14:366 Axiom cannot compute this integral
aa:=integrate(sin(a*x)^n,x)
--R
--R
@@ -731,7 +1899,7 @@ $$
<<*>>=
)clear all
---S 29 of 30
+--S 102 14:367 Axiom cannot compute this integral
aa:=integrate(1/(sin(a*x))^n,x)
--R
--R
@@ -739,7 +1907,7 @@ aa:=integrate(1/(sin(a*x))^n,x)
--R ++ 1
--I (1) | ---------- d%I
--R ++ n
---i sin(%I a)
+--I sin(%I a)
--R Type: Union(Expression
Integer,...)
--E
@
@@ -754,7 +1922,7 @@ $$
<<*>>=
)clear all
---S 30 of 30
+--S 103 14:368 Axiom cannot compute this integral
aa:=integrate(x/sin(a*x)^n,x)
--R
--R
diff --git a/src/input/schaum18.input.pamphlet
b/src/input/schaum18.input.pamphlet
index 3cb4766..95fceae 100644
--- a/src/input/schaum18.input.pamphlet
+++ b/src/input/schaum18.input.pamphlet
@@ -18,7 +18,7 @@ $$
)set message auto off
)clear all
---S 1 of 30
+--S 1
aa:=integrate(cos(a*x),x)
--R
--R
@@ -27,6 +27,22 @@ aa:=integrate(cos(a*x),x)
--R a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 2
+bb:=sin(a*x)/a
+--R
+--R sin(a x)
+--R (2) --------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 3 14:369 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.370~~~~~$\displaystyle
@@ -37,7 +53,7 @@ $$
<<*>>=
)clear all
---S 2 of 30
+--S 4
aa:=integrate(x*cos(a*x),x)
--R
--R
@@ -47,6 +63,23 @@ aa:=integrate(x*cos(a*x),x)
--R a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 5
+bb:=cos(a*x)/a^2+(x*sin(a*x))/a
+--R
+--R a x sin(a x) + cos(a x)
+--R (2) -----------------------
+--R 2
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 6 14:370 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.371~~~~~$\displaystyle
@@ -57,7 +90,7 @@ $$
<<*>>=
)clear all
---S 3 of 30
+--S 7
aa:=integrate(x^2*cos(a*x),x)
--R
--R
@@ -68,6 +101,24 @@ aa:=integrate(x^2*cos(a*x),x)
--R a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 8
+bb:=(2*x)/a^2*cos(a*x)+(x^2/a-2/a^3)*sin(a*x)
+--R
+--R 2 2
+--R (a x - 2)sin(a x) + 2a x cos(a x)
+--R (2) ----------------------------------
+--R 3
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 9 14:371 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.372~~~~~$\displaystyle
@@ -79,7 +130,7 @@ $$
<<*>>=
)clear all
---S 4 of 30
+--S 10
aa:=integrate(x^3*cos(a*x),x)
--R
--R
@@ -90,6 +141,24 @@ aa:=integrate(x^3*cos(a*x),x)
--R a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 11
+bb:=((3*x^2)/a^2-6/a^4)*cos(a*x)+(x^3/a-(6*x)/a^3)*sin(a*x)
+--R
+--R 3 3 2 2
+--R (a x - 6a x)sin(a x) + (3a x - 6)cos(a x)
+--R (2) -------------------------------------------
+--R 4
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 12 14:372 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.373~~~~~$\displaystyle
@@ -101,7 +170,7 @@ $$
<<*>>=
)clear all
---S 5 of 30
+--S 13 14:373 Schaums and Axiom agree by definition
aa:=integrate(cos(x)/x,x)
--R
--R
@@ -118,7 +187,7 @@ $$
<<*>>=
)clear all
---S 6 of 30
+--S 14 14:374 Axiom cannot compute this integral
aa:=integrate(cos(a*x)/x^2,x)
--R
--R
@@ -140,7 +209,7 @@ $$
<<*>>=
)clear all
---S 7 of 30
+--S 15
aa:=integrate(1/cos(a*x),x)
--R
--R
@@ -151,6 +220,59 @@ aa:=integrate(1/cos(a*x),x)
--R a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 16
+bb1:=1/a*log(sec(a*x)+tan(a*x))
+--R
+--R log(tan(a x) + sec(a x))
+--R (2) ------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 17
+bb2:=1/a*log(tan(%pi/4+(a*x)/2))
+--R
+--R 2a x + %pi
+--R log(tan(----------))
+--R 4
+--R (3) --------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 18
+cc1:=aa-bb1
+--R
+--R (4)
+--R sin(a x) + cos(a x) + 1
+--R - log(tan(a x) + sec(a x)) + log(-----------------------)
+--R cos(a x) + 1
+--R +
+--R sin(a x) - cos(a x) - 1
+--R - log(-----------------------)
+--R cos(a x) + 1
+--R /
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 19 14:375 Axiom cannot simplify these expressions
+cc2:=aa-bb2
+--R
+--R (5)
+--R 2a x + %pi sin(a x) + cos(a x) + 1
+--R - log(tan(----------)) + log(-----------------------)
+--R 4 cos(a x) + 1
+--R +
+--R sin(a x) - cos(a x) - 1
+--R - log(-----------------------)
+--R cos(a x) + 1
+--R /
+--R a
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.376~~~~~$\displaystyle
@@ -164,7 +286,7 @@ $$
<<*>>=
)clear all
---S 8 of 30
+--S 20 14:376 Axiom cannot compute this integral
aa:=integrate(x/cos(a*x),x)
--R
--R
@@ -184,7 +306,7 @@ $$
<<*>>=
)clear all
---S 9 of 30
+--S 21
aa:=integrate(cos(a*x)^2,x)
--R
--R
@@ -193,6 +315,40 @@ aa:=integrate(cos(a*x)^2,x)
--R 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 22
+bb:=x/2+sin(2*a*x)/(4*a)
+--R
+--R sin(2a x) + 2a x
+--R (2) ----------------
+--R 4a
+--R Type: Expression
Integer
+--E
+
+--S 23
+cc:=aa-bb
+--R
+--R - sin(2a x) + 2cos(a x)sin(a x)
+--R (3) -------------------------------
+--R 4a
+--R Type: Expression
Integer
+--E
+
+--S 24
+cossinrule:=rule(cos(b)*sin(a) == 1/2*(sin(a-b)+sin(a+b)))
+--R
+--I %S sin(b + a) - %S sin(b - a)
+--I (4) %S cos(b)sin(a) == -----------------------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 25 14:377 Schaums and Axiom agree
+dd:=cossinrule cc
+--R
+--R (5) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.378~~~~~$\displaystyle
@@ -203,7 +359,7 @@ $$
<<*>>=
)clear all
---S 10 of 30
+--S 26
aa:=integrate(x*cos(a*x)^2,x)
--R
--R
@@ -214,6 +370,87 @@ aa:=integrate(x*cos(a*x)^2,x)
--R 4a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 27
+bb:=x^2/4+(x*sin(2*a*x))/(4*a)+cos(2*a*x)/(8*a^2)
+--R
+--R 2 2
+--R 2a x sin(2a x) + cos(2a x) + 2a x
+--R (2) ----------------------------------
+--R 2
+--R 8a
+--R Type: Expression
Integer
+--E
+
+--S 28
+cc:=aa-bb
+--R
+--R 2
+--R - 2a x sin(2a x) + 4a x cos(a x)sin(a x) - cos(2a x) + 2cos(a x)
+--R (3) -----------------------------------------------------------------
+--R 2
+--R 8a
+--R Type: Expression
Integer
+--E
+
+--S 29
+cossinrule:=rule(cos(b)*sin(a) == 1/2*(sin(a-b)+sin(a+b)))
+--R
+--I %T sin(b + a) - %T sin(b - a)
+--I (4) %T cos(b)sin(a) == -----------------------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 30
+dd:=cossinrule cc
+--R
+--R 2
+--R - cos(2a x) + 2cos(a x)
+--R (5) ------------------------
+--R 2
+--R 8a
+--R Type: Expression
Integer
+--E
+
+--S 31
+coscosrule:=rule(cos(a)*cos(b) == 1/2*(cos(a-b)+cos(a+b)))
+--R
+--I %U cos(b + a) + %U cos(b - a)
+--I (6) %U cos(a)cos(b) == -----------------------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 32
+ee:=coscosrule dd
+--R
+--R 2
+--R - cos(2a x) + 2cos(a x)
+--R (7) ------------------------
+--R 2
+--R 8a
+--R Type: Expression
Integer
+--E
+
+--S 33
+cossqrrule1:=rule(cos(a)^2 == 1/2+1/2*cos(2*a))
+--R
+--R 2 cos(2a) + 1
+--R (8) cos(a) == -----------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 34 14:378 Schaums and Axiom differ by a constant
+ff:=cossqrrule1 ee
+--R
+--R 1
+--R (9) ---
+--R 2
+--R 8a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.379~~~~~$\displaystyle
@@ -224,7 +461,7 @@ $$
<<*>>=
)clear all
---S 11 of 30
+--S 35
aa:=integrate(cos(a*x)^3,x)
--R
--R
@@ -233,7 +470,42 @@ aa:=integrate(cos(a*x)^3,x)
--R (1) -----------------------
--R 3a
--R Type: Union(Expression
Integer,...)
+--E
+
+--S 36
+bb:=sin(a*x)/a-sin(a*x)^3/(3*a)
+--R
+--R 3
+--R - sin(a x) + 3sin(a x)
+--R (2) -----------------------
+--R 3a
+--R Type: Expression
Integer
--E
+
+--S 37
+cc:=aa-bb
+--R
+--R 3 2
+--R sin(a x) + (cos(a x) - 1)sin(a x)
+--R (3) -----------------------------------
+--R 3a
+--R Type: Expression
Integer
+--E
+
+--S 38
+cossqrrule:=rule(cos(a)^2 == 1-sin(a)^2)
+--R
+--R 2 2
+--R (4) cos(a) == - sin(a) + 1
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 39 14:379 Schaums and Axiom agree
+dd:=cossqrrule cc
+--R
+--R (5) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.380~~~~~$\displaystyle
@@ -244,7 +516,7 @@ $$
<<*>>=
)clear all
---S 12 of 30
+--S 40
aa:=integrate(cos(a*x)^4,x)
--R
--R
@@ -253,7 +525,26 @@ aa:=integrate(cos(a*x)^4,x)
--R (1) ---------------------------------------
--R 8a
--R Type: Union(Expression
Integer,...)
+--E
+
+--S 41
+bb:=(3*x)/8+sin(2*a*x)/(4*a)+sin(4*a*x)/(32*a)
+--R
+--R sin(4a x) + 8sin(2a x) + 12a x
+--R (2) ------------------------------
+--R 32a
+--R Type: Expression
Integer
--E
+
+--S 42 14:380 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R 3
+--R - sin(4a x) - 8sin(2a x) + (8cos(a x) + 12cos(a x))sin(a x)
+--R (3) ------------------------------------------------------------
+--R 32a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.381~~~~~$\displaystyle
@@ -264,7 +555,7 @@ $$
<<*>>=
)clear all
---S 13 of 30
+--S 43
aa:=integrate(1/cos(a*x)^2,x)
--R
--R
@@ -272,7 +563,41 @@ aa:=integrate(1/cos(a*x)^2,x)
--R (1) ----------
--R a cos(a x)
--R Type: Union(Expression
Integer,...)
+--E
+
+--S 44
+bb:=tan(a*x)/a
+--R
+--R tan(a x)
+--R (2) --------
+--R a
+--R Type: Expression
Integer
--E
+
+--S 45
+cc:=aa-bb
+--R
+--R - cos(a x)tan(a x) + sin(a x)
+--R (3) -----------------------------
+--R a cos(a x)
+--R Type: Expression
Integer
+--E
+
+--S 46
+tanrule:=rule(tan(a) == sin(a)/cos(a))
+--R
+--R sin(a)
+--R (4) tan(a) == ------
+--R cos(a)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 47 14:381 Schaums and Axiom agree
+dd:=tanrule cc
+--R
+--R (5) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.382~~~~~$\displaystyle
@@ -284,7 +609,7 @@ $$
<<*>>=
)clear all
---S 14 of 30
+--S 48
aa:=integrate(1/cos(a*x)^3,x)
--R
--R
@@ -300,7 +625,35 @@ aa:=integrate(1/cos(a*x)^3,x)
--R 2
--R 2a cos(a x)
--R Type: Union(Expression
Integer,...)
+--E
+
+--S 49
+bb:=sin(a*x)/(2*a*cos(a*x)^2)+1/(2*a)*log(tan(%pi/4+(a*x)/2))
+--R
+--R 2 2a x + %pi
+--R cos(a x) log(tan(----------)) + sin(a x)
+--R 4
+--R (2) ----------------------------------------
+--R 2
+--R 2a cos(a x)
+--R Type: Expression
Integer
--E
+
+--S 50 14:382 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R 2a x + %pi sin(a x) + cos(a x) + 1
+--R - log(tan(----------)) + log(-----------------------)
+--R 4 cos(a x) + 1
+--R +
+--R sin(a x) - cos(a x) - 1
+--R - log(-----------------------)
+--R cos(a x) + 1
+--R /
+--R 2a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.383~~~~~$\displaystyle
@@ -311,7 +664,7 @@ $$
<<*>>=
)clear all
---S 15 of 30
+--S 51
aa:=integrate(cos(p*x)*cos(q*x),x)
--R
--R
@@ -320,7 +673,35 @@ aa:=integrate(cos(p*x)*cos(q*x),x)
--R 2 2
--R q - p
--R Type: Union(Expression
Integer,...)
+--E
+
+--S 52
+bb:=(sin(a-p)*x)/(2*(a-p))+(sin(a+p)*x)/(2*(a+p))
+--R
+--R (p - a)x sin(p + a) + (p + a)x sin(p - a)
+--R (2) -----------------------------------------
+--R 2 2
+--R 2p - 2a
+--R Type: Expression
Integer
--E
+
+--S 53 14:383 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R 2 2 3 2
+--R (2p - 2a )q cos(p x)sin(q x) + (- 2p + 2a p)cos(q x)sin(p x)
+--R +
+--R 2 3 2
+--R ((- p + a)q + p - a p )x sin(p + a)
+--R +
+--R 2 3 2
+--R ((- p - a)q + p + a p )x sin(p - a)
+--R /
+--R 2 2 2 4 2 2
+--R (2p - 2a )q - 2p + 2a p
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.384~~~~~$\displaystyle
@@ -331,7 +712,7 @@ $$
<<*>>=
)clear all
---S 16 of 30
+--S 54
aa:=integrate(1/(1-cos(a*x)),x)
--R
--R
@@ -340,6 +721,24 @@ aa:=integrate(1/(1-cos(a*x)),x)
--R a sin(a x)
--R Type: Union(Expression
Integer,...)
--E
+
+--S 55
+bb:=-1/a*cot(a*x)/2
+--R
+--R cot(a x)
+--R (2) - --------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 56 14:384 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R cot(a x)sin(a x) - 2cos(a x) - 2
+--R (3) --------------------------------
+--R 2a sin(a x)
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.385~~~~~$\displaystyle
@@ -351,7 +750,7 @@ $$
<<*>>=
)clear all
---S 17 of 30
+--S 57
aa:=integrate(x/(1-cos(ax)),x)
--R
--R
@@ -361,6 +760,30 @@ aa:=integrate(x/(1-cos(ax)),x)
--R 2cos(ax) - 2
--R Type: Union(Expression
Integer,...)
--E
+
+--S 58
+bb:=-x/a*cot(a*x)/2+2/a^2*log(sin((a*x)/2))
+--R
+--R a x
+--R 4log(sin(---)) - a x cot(a x)
+--R 2
+--R (2) -----------------------------
+--R 2
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 59 14:385 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R a x 2 2
+--R (- 4cos(ax) + 4)log(sin(---)) + (a x cos(ax) - a x)cot(a x) - a x
+--R 2
+--R (3) ------------------------------------------------------------------
+--R 2 2
+--R 2a cos(ax) - 2a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.386~~~~~$\displaystyle
@@ -371,7 +794,7 @@ $$
<<*>>=
)clear all
---S 18 of 30
+--S 60
aa:=integrate(1/(1+cos(ax)),x)
--R
--R
@@ -379,7 +802,25 @@ aa:=integrate(1/(1+cos(ax)),x)
--R (1) -----------
--R cos(ax) + 1
--R Type: Union(Expression
Integer,...)
---E
+--E
+
+--S 61
+bb:=1/a*tan(a*x)/2
+--R
+--R tan(a x)
+--R (2) --------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 62 14:386 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (- cos(ax) - 1)tan(a x) + 2a x
+--R (3) ------------------------------
+--R 2a cos(ax) + 2a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.387~~~~~$\displaystyle
@@ -391,7 +832,7 @@ $$
<<*>>=
)clear all
---S 19 of 30
+--S 63
aa:=integrate(x/(1+cos(a*x)),x)
--R
--R
@@ -403,6 +844,35 @@ aa:=integrate(x/(1+cos(a*x)),x)
--R a cos(a x) + a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 64
+bb:=x/a*tan((a*x)/2)+2/a^2*log(cos((a*x)/2))
+--R
+--R a x a x
+--R 2log(cos(---)) + a x tan(---)
+--R 2 2
+--R (2) -----------------------------
+--R 2
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 65 14:387 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R a x 2
+--R (- 2cos(a x) - 2)log(cos(---)) + (- cos(a x) - 1)log(------------)
+--R 2 cos(a x) + 1
+--R +
+--R a x
+--R (- a x cos(a x) - a x)tan(---) + a x sin(a x)
+--R 2
+--R /
+--R 2 2
+--R a cos(a x) + a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.388~~~~~$\displaystyle
@@ -414,7 +884,7 @@ $$
<<*>>=
)clear all
---S 20 of 30
+--S 66
aa:=integrate(1/(1-cos(a*x))^2,x)
--R
--R
@@ -423,7 +893,32 @@ aa:=integrate(1/(1-cos(a*x))^2,x)
--R (1) --------------------------
--R (3a cos(a x) - 3a)sin(a x)
--R Type: Union(Expression
Integer,...)
+--E
+
+--S 67
+bb:=-1/(2*a)*cot((a*x)/2)-1/(6*a)*cot((a*x)/2)^3
+--R
+--R a x 3 a x
+--R - cot(---) - 3cot(---)
+--R 2 2
+--R (2) -----------------------
+--R 6a
+--R Type: Expression
Integer
--E
+
+--S 68 14:388 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R a x 3 a x
2
+--R ((cos(a x) - 1)cot(---) + (3cos(a x) - 3)cot(---))sin(a x) - 2cos(a
x)
+--R 2 2
+--R +
+--R 2cos(a x) + 4
+--R /
+--R (6a cos(a x) - 6a)sin(a x)
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.389~~~~~$\displaystyle
@@ -435,7 +930,7 @@ $$
<<*>>=
)clear all
---S 21 of 30
+--S 69
aa:=integrate(1/(1+cos(a*x))^2,x)
--R
--R
@@ -445,6 +940,34 @@ aa:=integrate(1/(1+cos(a*x))^2,x)
--R 3a cos(a x) + 6a cos(a x) + 3a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 70
+bb:=1/(2*a)*tan((a*x)/2)+1/(6*a)*tan((a*x)/2)^2
+--R
+--R a x 2 a x
+--R tan(---) + 3tan(---)
+--R 2 2
+--R (2) ---------------------
+--R 6a
+--R Type: Expression
Integer
+--E
+
+--S 71 14:389 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R 2 a x 2
+--R (- cos(a x) - 2cos(a x) - 1)tan(---)
+--R 2
+--R +
+--R 2 a x
+--R (- 3cos(a x) - 6cos(a x) - 3)tan(---) + (2cos(a x) + 4)sin(a x)
+--R 2
+--R /
+--R 2
+--R 6a cos(a x) + 12a cos(a x) + 6a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.390~~~~~$\displaystyle
@@ -466,7 +989,7 @@ $$
<<*>>=
)clear all
---S 22 of 30
+--S 72
aa:=integrate(1/(p+q*cos(a*x)),x)
--R
--R
@@ -491,6 +1014,128 @@ aa:=integrate(1/(p+q*cos(a*x)),x)
--R a\|- q + p
--R Type: Union(List Expression
Integer,...)
--E
+
+--S 73
+bb1:=2/(a*sqrt(p^2-q^2))*atan(sqrt((p-q)/(p+q)))*tan(1/2*a*x)
+--R
+--R +-------+
+--R a x |- q + p
+--R 2tan(---)atan( |------- )
+--R 2 \| q + p
+--R (2) -------------------------
+--R +---------+
+--R | 2 2
+--R a\|- q + p
+--R Type: Expression
Integer
+--E
+
+--S 74
+bb2:=a/(a*sqrt(q^2-p^2))*log((tan(1/2*a*x)+sqrt((q+p)/(q-p)))/(tan(1/2*a*x)-sqrt((q+p)/(q-p))))
+--R
+--R +-----+
+--R |q + p a x
+--R - |----- - tan(---)
+--R \|q - p 2
+--R log(---------------------)
+--R +-----+
+--R |q + p a x
+--R |----- - tan(---)
+--R \|q - p 2
+--R (3) --------------------------
+--R +-------+
+--R | 2 2
+--R \|q - p
+--R Type: Expression
Integer
+--E
+
+--S 75
+cc1:=aa.1-bb1
+--R
+--R (4)
+--R +-------+
+--R +---------+ | 2 2 2 2
+--R | 2 2 (- p cos(a x) - q)\|q - p + (- q + p )sin(a x)
+--R \|- q + p log(--------------------------------------------------)
+--R q cos(a x) + p
+--R +
+--R +-------+ +-------+
+--R a x | 2 2 |- q + p
+--R - 2tan(---)\|q - p atan( |------- )
+--R 2 \| q + p
+--R /
+--R +---------+ +-------+
+--R | 2 2 | 2 2
+--R a\|- q + p \|q - p
+--R Type: Expression
Integer
+--E
+
+--S 76
+cc2:=aa.2-bb1
+--R
+--R +---------+
+--R +-------+ | 2 2
+--R a x |- q + p sin(a x)\|- q + p
+--R - 2tan(---)atan( |------- ) + 2atan(-----------------------)
+--R 2 \| q + p (q + p)cos(a x) + q + p
+--R (5) ------------------------------------------------------------
+--R +---------+
+--R | 2 2
+--R a\|- q + p
+--R Type: Expression
Integer
+--E
+
+--S 77
+cc3:=aa.1-bb2
+--R
+--R (6)
+--R +-----+
+--R |q + p a x
+--R - |----- - tan(---)
+--R \|q - p 2
+--R - a log(---------------------)
+--R +-----+
+--R |q + p a x
+--R |----- - tan(---)
+--R \|q - p 2
+--R +
+--R +-------+
+--R | 2 2 2 2
+--R (- p cos(a x) - q)\|q - p + (- q + p )sin(a x)
+--R log(--------------------------------------------------)
+--R q cos(a x) + p
+--R /
+--R +-------+
+--R | 2 2
+--R a\|q - p
+--R Type: Expression
Integer
+--E
+
+--S 78 14:390 Axiom cannot simplify these expressions
+cc4:=aa.2-bb2
+--R
+--R (7)
+--R +-----+
+--R |q + p a x
+--R +---------+ - |----- - tan(---)
+--R | 2 2 \|q - p 2
+--R - a\|- q + p log(---------------------)
+--R +-----+
+--R |q + p a x
+--R |----- - tan(---)
+--R \|q - p 2
+--R +
+--R +---------+
+--R +-------+ | 2 2
+--R | 2 2 sin(a x)\|- q + p
+--R 2\|q - p atan(-----------------------)
+--R (q + p)cos(a x) + q + p
+--R /
+--R +---------+ +-------+
+--R | 2 2 | 2 2
+--R a\|- q + p \|q - p
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.391~~~~~$\displaystyle
@@ -502,7 +1147,7 @@ $$
<<*>>=
)clear all
---S 23 of 30
+--S 79
aa:=integrate(1/(p+q*cos(a*x))^2,x)
--R
--R
@@ -542,6 +1187,143 @@ aa:=integrate(1/(p+q*cos(a*x))^2,x)
--R ]
--R Type: Union(List Expression
Integer,...)
--E
+
+--S 80
+t1:=integrate(1/(p+q*cos(a*x)),x)
+--R
+--R (2)
+--R +-------+
+--R | 2 2 2 2
+--R (- p cos(a x) - q)\|q - p + (- q + p )sin(a x)
+--R log(--------------------------------------------------)
+--R q cos(a x) + p
+--R [-------------------------------------------------------,
+--R +-------+
+--R | 2 2
+--R a\|q - p
+--R +---------+
+--R | 2 2
+--R sin(a x)\|- q + p
+--R 2atan(-----------------------)
+--R (q + p)cos(a x) + q + p
+--R ------------------------------]
+--R +---------+
+--R | 2 2
+--R a\|- q + p
+--R Type: Union(List Expression
Integer,...)
+--E
+
+--S 81
+bb1:=(q*sin(a*x))/(a*(q^2-p^2)*(p+q*cos(a*x)))-p/(q^2-p^2)*t1.1
+--R
+--R (3)
+--R 2
+--R (- p q cos(a x) - p )
+--R *
+--R +-------+
+--R | 2 2 2 2
+--R (- p cos(a x) - q)\|q - p + (- q + p )sin(a x)
+--R log(--------------------------------------------------)
+--R q cos(a x) + p
+--R +
+--R +-------+
+--R | 2 2
+--R q sin(a x)\|q - p
+--R /
+--R +-------+
+--R 3 2 2 3 | 2 2
+--R ((a q - a p q)cos(a x) + a p q - a p )\|q - p
+--R Type: Expression
Integer
+--E
+
+--S 82
+bb2:=(q*sin(a*x))/(a*(q^2-p^2)*(p+q*cos(a*x)))-p/(q^2-p^2)*t1.2
+--R
+--R (4)
+--R +---------+
+--R | 2 2
+---------+
+--R 2 sin(a x)\|- q + p | 2
2
+--R (- 2p q cos(a x) - 2p )atan(-----------------------) + q sin(a x)\|- q
+ p
+--R (q + p)cos(a x) + q + p
+--R
-----------------------------------------------------------------------------
+--R +---------+
+--R 3 2 2 3 | 2 2
+--R ((a q - a p q)cos(a x) + a p q - a p )\|- q + p
+--R Type: Expression
Integer
+--E
+
+--S 83
+cc1:=aa.1-bb1
+--R
+--R (5)
+--R +-------+
+--R | 2 2 2 2
+--R (- p cos(a x) - q)\|q - p + (q - p )sin(a x)
+--R p log(------------------------------------------------)
+--R q cos(a x) + p
+--R +
+--R +-------+
+--R | 2 2 2 2
+--R (- p cos(a x) - q)\|q - p + (- q + p )sin(a x)
+--R p log(--------------------------------------------------)
+--R q cos(a x) + p
+--R /
+--R +-------+
+--R 2 2 | 2 2
+--R (a q - a p )\|q - p
+--R Type: Expression
Integer
+--E
+
+--S 84
+cc2:=aa.2-bb1
+--R
+--R (6)
+--R +-------+
+--R +---------+ | 2 2 2 2
+--R | 2 2 (- p cos(a x) - q)\|q - p + (- q + p )sin(a x)
+--R p\|- q + p log(--------------------------------------------------)
+--R q cos(a x) + p
+--R +
+--R +---------+
+--R +-------+ | 2 2
+--R | 2 2 sin(a x)\|- q + p
+--R - 2p\|q - p atan(-----------------------)
+--R (q + p)cos(a x) + q + p
+--R /
+--R +---------+ +-------+
+--R 2 2 | 2 2 | 2 2
+--R (a q - a p )\|- q + p \|q - p
+--R Type: Expression
Integer
+--E
+
+--S 85
+cc3:=aa.1-bb2
+--R
+--R (7)
+--R +-------+
+--R +---------+ | 2 2 2 2
+--R | 2 2 (- p cos(a x) - q)\|q - p + (q - p )sin(a x)
+--R p\|- q + p log(------------------------------------------------)
+--R q cos(a x) + p
+--R +
+--R +---------+
+--R +-------+ | 2 2
+--R | 2 2 sin(a x)\|- q + p
+--R 2p\|q - p atan(-----------------------)
+--R (q + p)cos(a x) + q + p
+--R /
+--R +---------+ +-------+
+--R 2 2 | 2 2 | 2 2
+--R (a q - a p )\|- q + p \|q - p
+--R Type: Expression
Integer
+--E
+
+--S 86 14:391 Schaums and Axiom agree
+cc4:=aa.2-bb2
+--R
+--R (8) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.392~~~~~$\displaystyle
@@ -552,7 +1334,7 @@ $$
<<*>>=
)clear all
---S 24 of 30
+--S 87
aa:=integrate(1/(p^2+a^2*cos(a*x)),x)
--R
--R
@@ -579,6 +1361,63 @@ aa:=integrate(1/(p^2+a^2*cos(a*x)),x)
--R a\|p - a
--R Type: Union(List Expression
Integer,...)
--E
+
+--S 88
+bb:=1/(a*p*sqrt(p^2+q^2))*atan((p*tan(a*x))/sqrt(p^2+q^2))
+--R
+--R p tan(a x)
+--R atan(----------)
+--R +-------+
+--R | 2 2
+--R \|q + p
+--R (2) ----------------
+--R +-------+
+--R | 2 2
+--R a p\|q + p
+--R Type: Expression
Integer
+--E
+
+--S 89
+cc1:=aa.1-bb
+--R
+--R (3)
+--R +---------+
+--R +-------+ 2 2 | 4 4 4 4
+--R | 2 2 (- p cos(a x) - a )\|- p + a + (p - a )sin(a x)
+--R p\|q + p log(---------------------------------------------------)
+--R 2 2
+--R a cos(a x) + p
+--R +
+--R +---------+
+--R | 4 4 p tan(a x)
+--R - \|- p + a atan(----------)
+--R +-------+
+--R | 2 2
+--R \|q + p
+--R /
+--R +---------+ +-------+
+--R | 4 4 | 2 2
+--R a p\|- p + a \|q + p
+--R Type: Expression
Integer
+--E
+
+--S 90 14:392 Axiom cannot simplify these expressions
+cc2:=aa.2-bb
+--R
+--R (4)
+--R +-------+
+--R +-------+ | 4 4 +-------+
+--R | 2 2 sin(a x)\|p - a | 4 4 p tan(a x)
+--R 2p\|q + p atan(---------------------------) - \|p - a
atan(----------)
+--R 2 2 2 2 +-------+
+--R (p + a )cos(a x) + p + a | 2 2
+--R \|q + p
+--R
--------------------------------------------------------------------------
+--R +-------+ +-------+
+--R | 4 4 | 2 2
+--R a p\|p - a \|q + p
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.393~~~~~$\displaystyle
@@ -598,7 +1437,7 @@ $$
<<*>>=
)clear all
---S 25 of 30
+--S 91
aa:=integrate(1/(p^2-q^2*cos(a*x)^2),x)
--R
--R
@@ -633,6 +1472,152 @@ aa:=integrate(1/(p^2-q^2*cos(a*x)^2),x)
--R ]
--R Type: Union(List Expression
Integer,...)
--E
+
+--S 92
+bb1:=1/(a*p*sqrt(p^2-a^2))*atan((p*tan(a*x))/sqrt(p^2-q^2))
+--R
+--R p tan(a x)
+--R atan(------------)
+--R +---------+
+--R | 2 2
+--R \|- q + p
+--R (2) ------------------
+--R +-------+
+--R | 2 2
+--R a p\|p - a
+--R Type: Expression
Integer
+--E
+
+--S 93
+bb2:=1/(2*a*p*sqrt(q^2-p^2))*log((p*tan(a*x)-sqrt(q^2-p^2))/(p*tan(a*x)+sqrt(q^2-p^2)))
+--R
+--R +-------+
+--R | 2 2
+--R - \|q - p + p tan(a x)
+--R log(-------------------------)
+--R +-------+
+--R | 2 2
+--R \|q - p + p tan(a x)
+--R (3) ------------------------------
+--R +-------+
+--R | 2 2
+--R 2a p\|q - p
+--R Type: Expression
Integer
+--E
+
+--S 94
+cc1:=aa.1-bb1
+--R
+--R (4)
+--R +-------+
+--R | 2 2
+--R \|p - a
+--R *
+--R log
+--R +-------+
+--R 2 2 2 2 | 2 2
+--R ((q - 2p )cos(a x) + p )\|q - p
+--R +
+--R 2 3
+--R (- 2p q + 2p )cos(a x)sin(a x)
+--R /
+--R 2 2 2
+--R q cos(a x) - p
+--R +
+--R +-------+
+--R | 2 2 p tan(a x)
+--R - 2\|q - p atan(------------)
+--R +---------+
+--R | 2 2
+--R \|- q + p
+--R /
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R 2a p\|p - a \|q - p
+--R Type: Expression
Integer
+--E
+
+--S 95
+cc2:=aa.2-bb1
+--R
+--R (5)
+--R +---------+
+--R +-------+ | 2 2 +---------+
+--R | 2 2 sin(a x)\|- q + p | 2 2 p tan(a x)
+--R \|p - a atan(--------------------) - \|- q + p atan(------------)
+--R 2p cos(a x) + 2p +---------+
+--R | 2 2
+--R \|- q + p
+--R +
+--R +-------+ 2 2 2
+--R | 2 2 ((q + p )cos(a x) + 2p )sin(a x)
+--R \|p - a atan(-------------------------------------------)
+--R +---------+
+--R 2 | 2 2
+--R (p cos(a x) + 2p cos(a x) + p)\|- q + p
+--R /
+--R +---------+ +-------+
+--R | 2 2 | 2 2
+--R a p\|- q + p \|p - a
+--R Type: Expression
Integer
+--E
+
+--S 96
+cc3:=aa.1-bb2
+--R
+--R (6)
+--R log
+--R +-------+
+--R 2 2 2 2 | 2 2 2 3
+--R ((q - 2p )cos(a x) + p )\|q - p + (- 2p q + 2p )cos(a
x)sin(a x)
+--R
----------------------------------------------------------------------
+--R 2 2 2
+--R q cos(a x) - p
+--R +
+--R +-------+
+--R | 2 2
+--R - \|q - p + p tan(a x)
+--R - log(-------------------------)
+--R +-------+
+--R | 2 2
+--R \|q - p + p tan(a x)
+--R /
+--R +-------+
+--R | 2 2
+--R 2a p\|q - p
+--R Type: Expression
Integer
+--E
+
+--S 97 14:393 Axiom cannot simplify these expressions
+cc4:=aa.2-bb2
+--R
+--R (7)
+--R +-------+
+--R +---------+ | 2 2
+--R | 2 2 - \|q - p + p tan(a x)
+--R - \|- q + p log(-------------------------)
+--R +-------+
+--R | 2 2
+--R \|q - p + p tan(a x)
+--R +
+--R +---------+
+--R +-------+ | 2 2
+--R | 2 2 sin(a x)\|- q + p
+--R 2\|q - p atan(--------------------)
+--R 2p cos(a x) + 2p
+--R +
+--R +-------+ 2 2 2
+--R | 2 2 ((q + p )cos(a x) + 2p )sin(a x)
+--R 2\|q - p atan(-------------------------------------------)
+--R +---------+
+--R 2 | 2 2
+--R (p cos(a x) + 2p cos(a x) + p)\|- q + p
+--R /
+--R +---------+ +-------+
+--R | 2 2 | 2 2
+--R 2a p\|- q + p \|q - p
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.394~~~~~$\displaystyle
@@ -644,7 +1629,7 @@ $$
<<*>>=
)clear all
---S 26 of 30
+--S 98 14:394 Axiom cannot compute this integral
aa:=integrate(x^m*cos(a*x),x)
--R
--R
@@ -664,7 +1649,7 @@ $$
<<*>>=
)clear all
---S 26 of 30
+--S 99 14:395 Axiom cannot compute this integral
aa:=integrate(cos(a*x)/x^n,x)
--R
--R
@@ -685,7 +1670,7 @@ $$
<<*>>=
)clear all
---S 28 of 30
+--S 100 14:396 Axiom cannot compute this integral
aa:=integrate(cos(a*x)^n,x)
--R
--R
@@ -706,7 +1691,7 @@ $$
<<*>>=
)clear all
---S 29 of 30
+--S 101 14:397 Axiom cannot compute this integral
aa:=integrate(1/(cos(a*x))^n,x)
--R
--R
@@ -729,7 +1714,7 @@ $$
<<*>>=
)clear all
---S 30 of 30
+--S 102 14:398 Axiom cannot compute this integral
aa:=integrate(x/cos(a*x)^n,x)
--R
--R
diff --git a/src/input/schaum19.input.pamphlet
b/src/input/schaum19.input.pamphlet
index 37e89ee..bf580f0 100644
--- a/src/input/schaum19.input.pamphlet
+++ b/src/input/schaum19.input.pamphlet
@@ -18,7 +18,7 @@ $$
)set message auto off
)clear all
---S 1 of 36
+--S 1
aa:=integrate(sin(a*x)*cos(a*x),x)
--R
--R
@@ -28,6 +28,43 @@ aa:=integrate(sin(a*x)*cos(a*x),x)
--R 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 2
+bb:=sin(a*x)^2/(2*a)
+--R
+--R 2
+--R sin(a x)
+--R (2) ---------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 3
+cc:=aa-bb
+--R
+--R 2 2
+--R - sin(a x) - cos(a x)
+--R (3) -----------------------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 4
+cossqrrule:=rule(cos(a)^2 == 1-sin(a)^2)
+--R
+--R 2 2
+--R (4) cos(a) == - sin(a) + 1
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 5 14:399 Schaums and Axiom differ by a constant
+dd:=cossqrrule cc
+--R
+--R 1
+--R (5) - --
+--R 2a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.400~~~~~$\displaystyle
@@ -38,7 +75,7 @@ $$
<<*>>=
)clear all
---S 2 of 36
+--S 6
aa:=integrate(sin(p*x)*cos(q*x),x)
--R
--R
@@ -48,6 +85,29 @@ aa:=integrate(sin(p*x)*cos(q*x),x)
--R q - p
--R Type: Union(Expression
Integer,...)
--E
+
+--S 7
+bb:=-(cos(p-q)*x)/(2*(p-q))-(cos(p+q)*x)/(2*(p+q))
+--R
+--R (- q + p)x cos(q + p) + (q + p)x cos(q - p)
+--R (2) -------------------------------------------
+--R 2 2
+--R 2q - 2p
+--R Type: Expression
Integer
+--E
+
+--S 8 14:400 Axiom is unable to simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R 2q sin(p x)sin(q x) + 2p cos(p x)cos(q x) + (q - p)x cos(q + p)
+--R +
+--R (- q - p)x cos(q - p)
+--R /
+--R 2 2
+--R 2q - 2p
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.401~~~~~$\displaystyle
@@ -58,7 +118,7 @@ $$
<<*>>=
)clear all
---S 3 of 36
+--S 9
aa:=integrate(sin(a*x)^n*cos(a*x),x)
--R
--R
@@ -68,6 +128,51 @@ aa:=integrate(sin(a*x)^n*cos(a*x),x)
--R a n + a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 10
+bb:=sin(a*x)^(n+1)/((n+1)*a)
+--R
+--R n + 1
+--R sin(a x)
+--R (2) -------------
+--R a n + a
+--R Type: Expression
Integer
+--E
+
+--S 11
+cc:=aa-bb
+--R
+--R n log(sin(a x)) n + 1
+--R sin(a x)%e - sin(a x)
+--R (3) -----------------------------------------
+--R a n + a
+--R Type: Expression
Integer
+--E
+
+--S 12
+explog:=rule(%e^(n*log(x)) == x^n)
+--R
+--R n log(x) n
+--R (4) %e == x
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 13
+dd:=explog cc
+--R
+--R n + 1 n
+--R - sin(a x) + sin(a x)sin(a x)
+--R (5) -----------------------------------
+--R a n + a
+--R Type: Expression
Integer
+--E
+
+--S 14 14:401 Schaums and Axiom agree
+ee:=complexNormalize dd
+--R
+--R (6) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.402~~~~~$\displaystyle
@@ -78,7 +183,7 @@ $$
<<*>>=
)clear all
---S 4 of 36
+--S 15
aa:=integrate(cos(a*x)^n*sin(a*x),x)
--R
--R
@@ -87,7 +192,52 @@ aa:=integrate(cos(a*x)^n*sin(a*x),x)
--R (1) - -------------------------
--R a n + a
--R Type: Union(Expression
Integer,...)
+--E
+
+--S 16
+bb:=-cos(a*x)^(n+1)/((n+1)*a)
+--R
+--R n + 1
+--R cos(a x)
+--R (2) - -------------
+--R a n + a
+--R Type: Expression
Integer
--E
+
+--S 17
+cc:=aa-bb
+--R
+--R n log(cos(a x)) n + 1
+--R - cos(a x)%e + cos(a x)
+--R (3) -------------------------------------------
+--R a n + a
+--R Type: Expression
Integer
+--E
+
+--S 18
+explog:=rule(%e^(n*log(x)) == x^n)
+--R
+--R n log(x) n
+--R (4) %e == x
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 19
+dd:=explog cc
+--R
+--R n + 1 n
+--R cos(a x) - cos(a x)cos(a x)
+--R (5) ---------------------------------
+--R a n + a
+--R Type: Expression
Integer
+--E
+
+--S 20 14:402 Schaums and Axiom agree
+ee:=complexNormalize dd
+--R
+--R (6) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.403~~~~~$\displaystyle
@@ -98,7 +248,7 @@ $$
<<*>>=
)clear all
---S 5 of 36
+--S 21
aa:=integrate(sin(a*x)^2*cos(a*x)^2,x)
--R
--R
@@ -108,6 +258,32 @@ aa:=integrate(sin(a*x)^2*cos(a*x)^2,x)
--R 8a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 22
+bb:=x/8-sin(4*a*x)/(32*a)
+--R
+--R - sin(4a x) + 4a x
+--R (2) ------------------
+--R 32a
+--R Type: Expression
Integer
+--E
+
+--S 23
+cc:=aa-bb
+--R
+--R 3
+--R sin(4a x) + (- 8cos(a x) + 4cos(a x))sin(a x)
+--R (3) ----------------------------------------------
+--R 32a
+--R Type: Expression
Integer
+--E
+
+--S 24 14:403 Schaums and Axiom agree
+dd:=complexNormalize cc
+--R
+--R (4) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.404~~~~~$\displaystyle
@@ -118,7 +294,7 @@ $$
<<*>>=
)clear all
---S 6 of 36
+--S 25
aa:=integrate(1/(sin(a*x)*cos(a*x)),x)
--R
--R
@@ -129,6 +305,55 @@ aa:=integrate(1/(sin(a*x)*cos(a*x)),x)
--R a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 26
+bb:=1/a*log(tan(a*x))
+--R
+--R log(tan(a x))
+--R (2) -------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 27
+cc:=aa-bb
+--R
+--R sin(a x) 2cos(a x)
+--R - log(tan(a x)) + log(------------) - log(- ------------)
+--R cos(a x) + 1 cos(a x) + 1
+--R (3) ---------------------------------------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 28
+tanrule:=rule(tan(a) == sin(a)/cos(a))
+--R
+--R sin(a)
+--R (4) tan(a) == ------
+--R cos(a)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 29
+dd:=tanrule cc
+--R
+--R sin(a x) sin(a x) 2cos(a x)
+--R - log(--------) + log(------------) - log(- ------------)
+--R cos(a x) cos(a x) + 1 cos(a x) + 1
+--R (5) ---------------------------------------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 30 14:404 Schaums and Axiom differ by a constant
+ee:=expandLog dd
+--R
+--R log(- 2)
+--R (6) - --------
+--R a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.405~~~~~$\displaystyle
@@ -139,7 +364,7 @@ $$
<<*>>=
)clear all
---S 7 of 36
+--S 31
aa:=integrate(1/(sin(a*x)^2*cos(a*x)),x)
--R
--R
@@ -155,6 +380,85 @@ aa:=integrate(1/(sin(a*x)^2*cos(a*x)),x)
--R a sin(a x)
--R Type: Union(Expression
Integer,...)
--E
+
+--S 32
+bb:=1/a*log(tan(%pi/4+(a*x)/2))-1/(a*sin(a*x))
+--R
+--R 2a x + %pi
+--R sin(a x)log(tan(----------)) - 1
+--R 4
+--R (2) --------------------------------
+--R a sin(a x)
+--R Type: Expression
Integer
+--E
+
+--S 33
+cc:=aa-bb
+--R
+--R (3)
+--R 2a x + %pi sin(a x) + cos(a x) + 1
+--R - log(tan(----------)) + log(-----------------------)
+--R 4 cos(a x) + 1
+--R +
+--R sin(a x) - cos(a x) - 1
+--R - log(-----------------------)
+--R cos(a x) + 1
+--R /
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 34
+tanrule:=rule(tan(a) == sin(a)/cos(a))
+--R
+--R sin(a)
+--R (4) tan(a) == ------
+--R cos(a)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 35
+dd:=tanrule cc
+--R
+--R (5)
+--R sin(a x) + cos(a x) + 1 sin(a x) - cos(a x) - 1
+--R log(-----------------------) - log(-----------------------)
+--R cos(a x) + 1 cos(a x) + 1
+--R +
+--R 2a x + %pi
+--R sin(----------)
+--R 4
+--R - log(---------------)
+--R 2a x + %pi
+--R cos(----------)
+--R 4
+--R /
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 36
+ee:=expandLog dd
+--R
+--R (6)
+--R log(sin(a x) + cos(a x) + 1) - log(sin(a x) - cos(a x) - 1)
+--R +
+--R 2a x + %pi 2a x + %pi
+--R - log(sin(----------)) + log(cos(----------))
+--R 4 4
+--R /
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 37 14:405 Schaums and Axiom differ by a constant
+ff:=complexNormalize %
+--R
+--R log(- 1)
+--R (7) --------
+--R a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.406~~~~~$\displaystyle
@@ -165,7 +469,7 @@ $$
<<*>>=
)clear all
---S 8 of 36
+--S 38
aa:=integrate(1/(sin(a*x)*cos(a*x)^2),x)
--R
--R
@@ -176,6 +480,72 @@ aa:=integrate(1/(sin(a*x)*cos(a*x)^2),x)
--R a cos(a x)
--R Type: Union(Expression
Integer,...)
--E
+
+--S 39
+bb:=1/a*log(tan((a*x)/2))+1/(a*cos(a*x))
+--R
+--R a x
+--R cos(a x)log(tan(---)) + 1
+--R 2
+--R (2) -------------------------
+--R a cos(a x)
+--R Type: Expression
Integer
+--E
+
+--S 40
+cc:=aa-bb
+--R
+--R a x sin(a x)
+--R - log(tan(---)) + log(------------) + 1
+--R 2 cos(a x) + 1
+--R (3) ---------------------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 41
+tanrule:=rule(tan(a) == sin(a)/cos(a))
+--R
+--R sin(a)
+--R (4) tan(a) == ------
+--R cos(a)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 42
+dd:=tanrule cc
+--R
+--R a x
+--R sin(---)
+--R sin(a x) 2
+--R log(------------) - log(--------) + 1
+--R cos(a x) + 1 a x
+--R cos(---)
+--R 2
+--R (5) -------------------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 43
+ee:=expandLog dd
+--R
+--R a x a x
+--R log(sin(a x)) - log(sin(---)) - log(cos(a x) + 1) + log(cos(---)) +
1
+--R 2 2
+--R (6)
---------------------------------------------------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 44 14:406 Schaums and Axiom differ by a constant
+ff:=complexNormalize ee
+--R
+--R 1
+--R (7) -
+--R a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.407~~~~~$\displaystyle
@@ -186,7 +556,7 @@ $$
<<*>>=
)clear all
---S 9 of 36
+--S 45
aa:=integrate(1/(sin(a*x)^2*cos(a*x)^2),x)
--R
--R
@@ -196,6 +566,51 @@ aa:=integrate(1/(sin(a*x)^2*cos(a*x)^2),x)
--R a cos(a x)sin(a x)
--R Type: Union(Expression
Integer,...)
--E
+
+--S 46
+bb:=-(2*cot(2*a*x))/a
+--R
+--R 2cot(2a x)
+--R (2) - ----------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 47
+cc:=aa-bb
+--R
+--R 2
+--R 2cos(a x)cot(2a x)sin(a x) - 2cos(a x) + 1
+--R (3) -------------------------------------------
+--R a cos(a x)sin(a x)
+--R Type: Expression
Integer
+--E
+
+--S 48
+cotrule:=rule(cot(a) == cos(a)/sin(a))
+--R
+--R cos(a)
+--R (4) cot(a) == ------
+--R sin(a)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 49
+dd:=cotrule cc
+--R
+--R 2
+--R (- 2cos(a x) + 1)sin(2a x) + 2cos(a x)cos(2a x)sin(a x)
+--R (5) --------------------------------------------------------
+--R a cos(a x)sin(a x)sin(2a x)
+--R Type: Expression
Integer
+--E
+
+--S 50 14:407 Schaums and Axiom agree
+ee:=complexNormalize dd
+--R
+--R (6) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.408~~~~~$\displaystyle
@@ -206,7 +621,7 @@ $$
<<*>>=
)clear all
---S 10 of 36
+--S 51
aa:=integrate(sin(a*x)^2/cos(a*x),x)
--R
--R
@@ -217,6 +632,85 @@ aa:=integrate(sin(a*x)^2/cos(a*x),x)
--R a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 52
+bb:=-sin(a*x)/a+1/a*log(tan((a*x)/2+%pi/4))
+--R
+--R 2a x + %pi
+--R log(tan(----------)) - sin(a x)
+--R 4
+--R (2) -------------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 53
+cc:=aa-bb
+--R
+--R (3)
+--R 2a x + %pi sin(a x) + cos(a x) + 1
+--R - log(tan(----------)) + log(-----------------------)
+--R 4 cos(a x) + 1
+--R +
+--R sin(a x) - cos(a x) - 1
+--R - log(-----------------------)
+--R cos(a x) + 1
+--R /
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 54
+tanrule:=rule(tan(a) == sin(a)/cos(a))
+--R
+--R sin(a)
+--R (4) tan(a) == ------
+--R cos(a)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 55
+dd:=tanrule cc
+--R
+--R (5)
+--R sin(a x) + cos(a x) + 1 sin(a x) - cos(a x) - 1
+--R log(-----------------------) - log(-----------------------)
+--R cos(a x) + 1 cos(a x) + 1
+--R +
+--R 2a x + %pi
+--R sin(----------)
+--R 4
+--R - log(---------------)
+--R 2a x + %pi
+--R cos(----------)
+--R 4
+--R /
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 56
+ee:=expandLog dd
+--R
+--R (6)
+--R log(sin(a x) + cos(a x) + 1) - log(sin(a x) - cos(a x) - 1)
+--R +
+--R 2a x + %pi 2a x + %pi
+--R - log(sin(----------)) + log(cos(----------))
+--R 4 4
+--R /
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 57 14:408 Schaums and Axiom differ by a constant
+ff:=complexNormalize ee
+--R
+--R log(- 1)
+--R (7) --------
+--R a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.409~~~~~$\displaystyle
@@ -227,7 +721,7 @@ $$
<<*>>=
)clear all
---S 11 of 36
+--S 58
aa:=integrate(cos(a*x)^2/sin(a*x),x)
--R
--R
@@ -238,6 +732,70 @@ aa:=integrate(cos(a*x)^2/sin(a*x),x)
--R a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 59
+bb:=cos(a*x)/a+1/a*log(tan((a*x)/2))
+--R
+--R a x
+--R log(tan(---)) + cos(a x)
+--R 2
+--R (2) ------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 60
+cc:=aa-bb
+--R
+--R a x sin(a x)
+--R - log(tan(---)) + log(------------)
+--R 2 cos(a x) + 1
+--R (3) -----------------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 61
+tanrule:=rule(tan(a) == sin(a)/cos(a))
+--R
+--R sin(a)
+--R (4) tan(a) == ------
+--R cos(a)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 62
+dd:=tanrule cc
+--R
+--R a x
+--R sin(---)
+--R sin(a x) 2
+--R log(------------) - log(--------)
+--R cos(a x) + 1 a x
+--R cos(---)
+--R 2
+--R (5) ---------------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 63
+ee:=expandLog dd
+--R
+--R a x a x
+--R log(sin(a x)) - log(sin(---)) - log(cos(a x) + 1) + log(cos(---))
+--R 2 2
+--R (6) -----------------------------------------------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 64 14:409 Schaums and Axiom agree
+ff:=complexNormalize ee
+--R
+--R (7) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.410~~~~~$\displaystyle
@@ -249,7 +807,7 @@ $$
<<*>>=
)clear all
---S 12 of 36
+--S 65
aa:=integrate(1/(cos(a*x)*(1+sin(a*x))),x)
--R
--R
@@ -266,9 +824,88 @@ aa:=integrate(1/(cos(a*x)*(1+sin(a*x))),x)
--R Type: Union(Expression
Integer,...)
--E
+--S 66
+bb:=-1/(2*a*(1+sin(a*x)))+1/(2*a)*log(tan((a*x)/2+%pi/4))
+--R
+--R 2a x + %pi
+--R (sin(a x) + 1)log(tan(----------)) - 1
+--R 4
+--R (2) --------------------------------------
+--R 2a sin(a x) + 2a
+--R Type: Expression
Integer
+--E
+
+--S 67
+cc:=aa-bb
+--R
+--R (3)
+--R 2a x + %pi sin(a x) + cos(a x) + 1
+--R - log(tan(----------)) + log(-----------------------)
+--R 4 cos(a x) + 1
+--R +
+--R sin(a x) - cos(a x) - 1
+--R - log(-----------------------) + 1
+--R cos(a x) + 1
+--R /
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 68
+tanrule:=rule(tan(a) == sin(a)/cos(a))
+--R
+--R sin(a)
+--R (4) tan(a) == ------
+--R cos(a)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 69
+dd:=tanrule cc
+--R
+--R (5)
+--R sin(a x) + cos(a x) + 1 sin(a x) - cos(a x) - 1
+--R log(-----------------------) - log(-----------------------)
+--R cos(a x) + 1 cos(a x) + 1
+--R +
+--R 2a x + %pi
+--R sin(----------)
+--R 4
+--R - log(---------------) + 1
+--R 2a x + %pi
+--R cos(----------)
+--R 4
+--R /
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 70
+ee:=expandLog dd
+--R
+--R (6)
+--R log(sin(a x) + cos(a x) + 1) - log(sin(a x) - cos(a x) - 1)
+--R +
+--R 2a x + %pi 2a x + %pi
+--R - log(sin(----------)) + log(cos(----------)) + 1
+--R 4 4
+--R /
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 71
+ff:=complexNormalize ee
+--R
+--R log(- 1) + 1
+--R (7) ------------
+--R 2a
+--R Type: Expression
Integer
+--E
+
)clear all
---S 13 of 36
+--S 72
aa:=integrate(1/(cos(a*x)*(1-sin(a*x))),x)
--R
--R
@@ -284,6 +921,86 @@ aa:=integrate(1/(cos(a*x)*(1-sin(a*x))),x)
--R 2a sin(a x) - 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 73
+bb:=1/(2*a*(1-sin(a*x)))+1/(2*a)*log(tan((a*x)/2+%pi/4))
+--R
+--R 2a x + %pi
+--R (sin(a x) - 1)log(tan(----------)) - 1
+--R 4
+--R (2) --------------------------------------
+--R 2a sin(a x) - 2a
+--R Type: Expression
Integer
+--E
+
+--S 74
+cc:=aa-bb
+--R
+--R (3)
+--R 2a x + %pi sin(a x) + cos(a x) + 1
+--R - log(tan(----------)) + log(-----------------------)
+--R 4 cos(a x) + 1
+--R +
+--R sin(a x) - cos(a x) - 1
+--R - log(-----------------------) - 1
+--R cos(a x) + 1
+--R /
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 75
+tanrule:=rule(tan(a) == sin(a)/cos(a))
+--R
+--R sin(a)
+--R (4) tan(a) == ------
+--R cos(a)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 76
+dd:=tanrule cc
+--R
+--R (5)
+--R sin(a x) + cos(a x) + 1 sin(a x) - cos(a x) - 1
+--R log(-----------------------) - log(-----------------------)
+--R cos(a x) + 1 cos(a x) + 1
+--R +
+--R 2a x + %pi
+--R sin(----------)
+--R 4
+--R - log(---------------) - 1
+--R 2a x + %pi
+--R cos(----------)
+--R 4
+--R /
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 77
+ee:=expandLog dd
+--R
+--R (6)
+--R log(sin(a x) + cos(a x) + 1) - log(sin(a x) - cos(a x) - 1)
+--R +
+--R 2a x + %pi 2a x + %pi
+--R - log(sin(----------)) + log(cos(----------)) - 1
+--R 4 4
+--R /
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 78 14:410 Schaums and Axiom differ by a constant
+ff:=complexNormalize ee
+--R
+--R log(- 1) - 1
+--R (7) ------------
+--R 2a
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.411~~~~~$\displaystyle
@@ -294,7 +1011,7 @@ $$
<<*>>=
)clear all
---S 14 of 36
+--S 79
aa:=integrate(1/(sin(a*x)*(1+cos(a*x))),x)
--R
--R
@@ -306,9 +1023,76 @@ aa:=integrate(1/(sin(a*x)*(1+cos(a*x))),x)
--R Type: Union(Expression
Integer,...)
--E
+--S 80
+bb:=1/(2*a*(1+cos(a*x)))+1/(2*a)*log(tan((a*x)/2))
+--R
+--R a x
+--R (cos(a x) + 1)log(tan(---)) + 1
+--R 2
+--R (2) -------------------------------
+--R 2a cos(a x) + 2a
+--R Type: Expression
Integer
+--E
+
+--S 81
+cc:=aa-bb
+--R
+--R a x sin(a x)
+--R - 2log(tan(---)) + 2log(------------) - 1
+--R 2 cos(a x) + 1
+--R (3) -----------------------------------------
+--R 4a
+--R Type: Expression
Integer
+--E
+
+--S 82
+tanrule:=rule(tan(a) == sin(a)/cos(a))
+--R
+--R sin(a)
+--R (4) tan(a) == ------
+--R cos(a)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 83
+dd:=tanrule cc
+--R
+--R a x
+--R sin(---)
+--R sin(a x) 2
+--R 2log(------------) - 2log(--------) - 1
+--R cos(a x) + 1 a x
+--R cos(---)
+--R 2
+--R (5) ---------------------------------------
+--R 4a
+--R Type: Expression
Integer
+--E
+
+--S 84
+ee:=expandLog dd
+--R
+--R (6)
+--R a x a x
+--R 2log(sin(a x)) - 2log(sin(---)) - 2log(cos(a x) + 1) + 2log(cos(---)) - 1
+--R 2 2
+--R -------------------------------------------------------------------------
+--R 4a
+--R Type: Expression
Integer
+--E
+
+--S 85
+ff:=complexNormalize ee
+--R
+--R 1
+--R (7) - --
+--R 4a
+--R Type: Expression
Integer
+--E
+
)clear all
---S 15 of 36
+--S 86
aa:=integrate(1/(sin(a*x)*(1-cos(a*x))),x)
--R
--R
@@ -319,6 +1103,73 @@ aa:=integrate(1/(sin(a*x)*(1-cos(a*x))),x)
--R 4a cos(a x) - 4a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 87
+bb:=-1/(2*a*(1-cos(a*x)))+1/(2*a)*log(tan((a*x)/2))
+--R
+--R a x
+--R (cos(a x) - 1)log(tan(---)) + 1
+--R 2
+--R (2) -------------------------------
+--R 2a cos(a x) - 2a
+--R Type: Expression
Integer
+--E
+
+--S 88
+cc:=aa-bb
+--R
+--R a x sin(a x)
+--R - 2log(tan(---)) + 2log(------------) + 1
+--R 2 cos(a x) + 1
+--R (3) -----------------------------------------
+--R 4a
+--R Type: Expression
Integer
+--E
+
+--S 89
+tanrule:=rule(tan(a) == sin(a)/cos(a))
+--R
+--R sin(a)
+--R (4) tan(a) == ------
+--R cos(a)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 90
+dd:=tanrule cc
+--R
+--R a x
+--R sin(---)
+--R sin(a x) 2
+--R 2log(------------) - 2log(--------) + 1
+--R cos(a x) + 1 a x
+--R cos(---)
+--R 2
+--R (5) ---------------------------------------
+--R 4a
+--R Type: Expression
Integer
+--E
+
+--S 91
+ee:=expandLog dd
+--R
+--R (6)
+--R a x a x
+--R 2log(sin(a x)) - 2log(sin(---)) - 2log(cos(a x) + 1) + 2log(cos(---)) + 1
+--R 2 2
+--R -------------------------------------------------------------------------
+--R 4a
+--R Type: Expression
Integer
+--E
+
+--S 92 14:411 Schaums and Axiom differ by a constant
+ff:=complexNormalize ee
+--R
+--R 1
+--R (7) --
+--R 4a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.412~~~~~$\displaystyle
@@ -329,7 +1180,7 @@ $$
<<*>>=
)clear all
---S 16 of 36
+--S 93
aa:=integrate(1/(sin(a*x)+cos(a*x)),x)
--R
--R
@@ -342,9 +1193,37 @@ aa:=integrate(1/(sin(a*x)+cos(a*x)),x)
--R Type: Union(Expression
Integer,...)
--E
+--S 94
+bb:=1/(a*sqrt(2))*log(tan((a*x)/2+%pi/8))
+--R
+--R +-+ 4a x + %pi
+--R \|2 log(tan(----------))
+--R 8
+--R (2) ------------------------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 95
+cc:=aa-bb
+--R
+--R (3)
+--R +-+ 4a x + %pi
+--R - \|2 log(tan(----------))
+--R 8
+--R +
+--R +-+ +-+ +-+
+--R +-+ (- \|2 + 1)sin(a x) + (\|2 - 1)cos(a x) + \|2 - 2
+--R \|2 log(----------------------------------------------------)
+--R sin(a x) + cos(a x)
+--R /
+--R 2a
+--R Type: Expression
Integer
+--E
+
)clear all
---S 17 of 36
+--S 96
aa:=integrate(1/(sin(a*x)-cos(a*x)),x)
--R
--R
@@ -356,6 +1235,35 @@ aa:=integrate(1/(sin(a*x)-cos(a*x)),x)
--R 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 97
+bb:=1/(a*sqrt(2))*log(tan((a*x)/2-%pi/8))
+--R
+--R +-+ 4a x - %pi
+--R \|2 log(tan(----------))
+--R 8
+--R (2) ------------------------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 98 14:412 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R +-+ 4a x - %pi
+--R - \|2 log(tan(----------))
+--R 8
+--R +
+--R +-+ +-+ +-+
+--R +-+ (- \|2 + 1)sin(a x) + (- \|2 + 1)cos(a x) - \|2 + 2
+--R \|2 log(------------------------------------------------------)
+--R sin(a x) - cos(a x)
+--R /
+--R 2a
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.413~~~~~$\displaystyle
@@ -366,7 +1274,7 @@ $$
<<*>>=
)clear all
---S 18 of 36
+--S 99
aa:=integrate(sin(a*x)/(sin(a*x)+cos(a*x)),x)
--R
--R
@@ -378,9 +1286,48 @@ aa:=integrate(sin(a*x)/(sin(a*x)+cos(a*x)),x)
--R Type: Union(Expression
Integer,...)
--E
+--S 100
+bb:=x/2-1/(2*a)*log(sin(a*x)+cos(a*x))
+--R
+--R - log(sin(a x) + cos(a x)) + a x
+--R (2) --------------------------------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 101
+cc:=aa-bb
+--R
+--R (3)
+--R 2 - 2sin(a x) - 2cos(a
x)
+--R log(sin(a x) + cos(a x)) + log(------------) -
log(-----------------------)
+--R cos(a x) + 1 cos(a x) + 1
+--R
---------------------------------------------------------------------------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 102
+dd:=expandLog cc
+--R
+--R log(sin(a x) + cos(a x)) - log(- sin(a x) - cos(a x))
+--R (4) -----------------------------------------------------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 103
+ee:=complexNormalize dd
+--R
+--R log(- 1)
+--R (5) --------
+--R 2a
+--R Type: Expression
Integer
+--E
+
)clear all
---S 19 of 36
+--S 104
aa:=integrate(sin(a*x)/(sin(a*x)-cos(a*x)),x)
--R
--R
@@ -391,6 +1338,34 @@ aa:=integrate(sin(a*x)/(sin(a*x)-cos(a*x)),x)
--R 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 105
+bb:=x/2+1/(2*a)*log(sin(a*x)-cos(a*x))
+--R
+--R log(sin(a x) - cos(a x)) + a x
+--R (2) ------------------------------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 106
+cc:=aa-bb
+--R
+--R (3)
+--R 2sin(a x) - 2cos(a x) 2
+--R - log(sin(a x) - cos(a x)) + log(---------------------) -
log(------------)
+--R cos(a x) + 1 cos(a x) +
1
+--R
---------------------------------------------------------------------------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 107 14:413 Schaums and Axiom agree
+dd:=expandLog cc
+--R
+--R (4) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.414~~~~~$\displaystyle
@@ -401,7 +1376,7 @@ $$
<<*>>=
)clear all
---S 20 of 36
+--S 108
aa:=integrate(cos(a*x)/(sin(a*x)+cos(a*x)),x)
--R
--R
@@ -413,9 +1388,48 @@ aa:=integrate(cos(a*x)/(sin(a*x)+cos(a*x)),x)
--R Type: Union(Expression
Integer,...)
--E
+--S 109
+bb:=x/2+1/(2*a)*log(sin(a*x)+cos(a*x))
+--R
+--R log(sin(a x) + cos(a x)) + a x
+--R (2) ------------------------------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 110
+cc:=aa-bb
+--R
+--R (3)
+--R 2 - 2sin(a x) -
2cos(a x)
+--R - log(sin(a x) + cos(a x)) - log(------------) +
log(-----------------------)
+--R cos(a x) + 1 cos(a x) + 1
+--R
-----------------------------------------------------------------------------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 111
+dd:=expandLog cc
+--R
+--R - log(sin(a x) + cos(a x)) + log(- sin(a x) - cos(a x))
+--R (4) -------------------------------------------------------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 112
+ee:=complexNormalize dd
+--R
+--R log(- 1)
+--R (5) - --------
+--R 2a
+--R Type: Expression
Integer
+--E
+
)clear all
---S 21 of 36
+--S 113
aa:=integrate(cos(a*x)/(sin(a*x)-cos(a*x)),x)
--R
--R
@@ -426,6 +1440,34 @@ aa:=integrate(cos(a*x)/(sin(a*x)-cos(a*x)),x)
--R 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 114
+bb:=-x/2+1/(2*a)*log(sin(a*x)-cos(a*x))
+--R
+--R log(sin(a x) - cos(a x)) - a x
+--R (2) ------------------------------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 115
+cc:=aa-bb
+--R
+--R (3)
+--R 2sin(a x) - 2cos(a x) 2
+--R - log(sin(a x) - cos(a x)) + log(---------------------) -
log(------------)
+--R cos(a x) + 1 cos(a x) +
1
+--R
---------------------------------------------------------------------------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 116 14:414 Schaums and Axiom agree
+dd:=expandLog cc
+--R
+--R (4) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.415~~~~~$\displaystyle
@@ -436,7 +1478,7 @@ $$
<<*>>=
)clear all
---S 22 of 36
+--S 117
aa:=integrate(sin(a*x)/(p+q*cos(a*x)),x)
--R
--R
@@ -447,6 +1489,44 @@ aa:=integrate(sin(a*x)/(p+q*cos(a*x)),x)
--R a q
--R Type: Union(Expression
Integer,...)
--E
+
+--S 118
+bb:=-1/(a*q)*log(p+q*cos(a*x))
+--R
+--R log(q cos(a x) + p)
+--R (2) - -------------------
+--R a q
+--R Type: Expression
Integer
+--E
+
+--S 119
+cc:=aa-bb
+--R
+--R 2 - 2q cos(a x) - 2p
+--R log(q cos(a x) + p) + log(------------) - log(------------------)
+--R cos(a x) + 1 cos(a x) + 1
+--R (3) -----------------------------------------------------------------
+--R a q
+--R Type: Expression
Integer
+--E
+
+--S 120
+dd:=expandLog cc
+--R
+--R log(q cos(a x) + p) - log(- q cos(a x) - p)
+--R (4) -------------------------------------------
+--R a q
+--R Type: Expression
Integer
+--E
+
+--S 121 14:415 Schaums and Axiom differ by a constant
+ee:=complexNormalize dd
+--R
+--R log(- 1)
+--R (5) --------
+--R a q
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.416~~~~~$\displaystyle
@@ -457,7 +1537,7 @@ $$
<<*>>=
)clear all
---S 23 of 36
+--S 122
aa:=integrate(cos(a*x)/(p+q*sin(a*x)),x)
--R
--R
@@ -468,6 +1548,33 @@ aa:=integrate(cos(a*x)/(p+q*sin(a*x)),x)
--R a q
--R Type: Union(Expression
Integer,...)
--E
+
+--S 123
+bb:=1/(a*q)*log(p+q*sin(a*x))
+--R
+--R log(q sin(a x) + p)
+--R (2) -------------------
+--R a q
+--R Type: Expression
Integer
+--E
+
+--S 124
+cc:=aa-bb
+--R
+--R 2q sin(a x) + 2p 2
+--R - log(q sin(a x) + p) + log(----------------) - log(------------)
+--R cos(a x) + 1 cos(a x) + 1
+--R (3) -----------------------------------------------------------------
+--R a q
+--R Type: Expression
Integer
+--E
+
+--S 125 14:416 Schaums and Axiom differ by a constant
+dd:=expandLog cc
+--R
+--R (4) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.417~~~~~$\displaystyle
@@ -478,7 +1585,7 @@ $$
<<*>>=
)clear all
---S 24 of 36
+--S 126
aa:=integrate(sin(a*x)/(p+q*cos(a*x))^n,x)
--R
--R
@@ -488,6 +1595,53 @@ aa:=integrate(sin(a*x)/(p+q*cos(a*x))^n,x)
--R (a n - a)q %e
--R Type: Union(Expression
Integer,...)
--E
+
+--S 127
+bb:=1/(a*q*(n-1)*(p+q*cos(a*x))^(n-1))
+--R
+--R 1
+--R (2) --------------------------------
+--R n - 1
+--R (a n - a)q (q cos(a x) + p)
+--R Type: Expression
Integer
+--E
+
+--S 128
+cc:=aa-bb
+--R
+--R n log(q cos(a x) + p) n - 1
+--R - %e + (q cos(a x) + p)(q cos(a x) + p)
+--R (3) -----------------------------------------------------------------
+--R n - 1 n log(q cos(a x) + p)
+--R (a n - a)q (q cos(a x) + p) %e
+--R Type: Expression
Integer
+--E
+
+--S 129
+explog:=rule(%e^(n*log(x)) == x^n)
+--R
+--R n log(x) n
+--R (4) %e == x
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 130
+dd:=explog cc
+--R
+--R n n - 1
+--R - (q cos(a x) + p) + (q cos(a x) + p)(q cos(a x) + p)
+--R (5) -----------------------------------------------------------
+--R n - 1 n
+--R (a n - a)q (q cos(a x) + p) (q cos(a x) + p)
+--R Type: Expression
Integer
+--E
+
+--S 131 14:417 Schaums and Axiom agree
+ee:=complexNormalize dd
+--R
+--R (6) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.418~~~~~$\displaystyle
@@ -498,7 +1652,7 @@ $$
<<*>>=
)clear all
---S 25 of 36
+--S 132
aa:=integrate(cos(a*x)/(p+q*sin(a*x))^n,x)
--R
--R
@@ -508,6 +1662,53 @@ aa:=integrate(cos(a*x)/(p+q*sin(a*x))^n,x)
--R (a n - a)q %e
--R Type: Union(Expression
Integer,...)
--E
+
+--S 133
+bb:=-1/(a*q*(n-1)*(p+q*sin(a*x))^(n-1))
+--R
+--R 1
+--R (2) - --------------------------------
+--R n - 1
+--R (a n - a)q (q sin(a x) + p)
+--R Type: Expression
Integer
+--E
+
+--S 134
+cc:=aa-bb
+--R
+--R n log(q sin(a x) + p) n - 1
+--R %e + (- q sin(a x) - p)(q sin(a x) + p)
+--R (3) -----------------------------------------------------------------
+--R n - 1 n log(q sin(a x) + p)
+--R (a n - a)q (q sin(a x) + p) %e
+--R Type: Expression
Integer
+--E
+
+--S 135
+explog:=rule(%e^(n*log(x)) == x^n)
+--R
+--R n log(x) n
+--R (4) %e == x
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 136
+dd:=explog cc
+--R
+--R n n - 1
+--R (q sin(a x) + p) + (- q sin(a x) - p)(q sin(a x) + p)
+--R (5) -----------------------------------------------------------
+--R n - 1 n
+--R (a n - a)q (q sin(a x) + p) (q sin(a x) + p)
+--R Type: Expression
Integer
+--E
+
+--S 137 14:418 Schaums and Axiom agree
+ee:=complexNormalize dd
+--R
+--R (6) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.419~~~~~$\displaystyle
@@ -518,7 +1719,7 @@ $$
<<*>>=
)clear all
---S 26 of 36
+--S 138
aa:=integrate(1/(p*sin(a*x)+q*cos(a*x)),x)
--R
--R
@@ -538,6 +1739,47 @@ aa:=integrate(1/(p*sin(a*x)+q*cos(a*x)),x)
--R a\|q + p
--R Type: Union(Expression
Integer,...)
--E
+
+--S 139
+bb:=1/(a*sqrt(p^2+q^2))*log(tan((a*x+atan(q/p))/2))
+--R
+--R q
+--R atan(-) + a x
+--R p
+--R log(tan(-------------))
+--R 2
+--R (2) -----------------------
+--R +-------+
+--R | 2 2
+--R a\|q + p
+--R Type: Expression
Integer
+--E
+
+--S 140 14:419 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R q
+--R atan(-) + a x
+--R p
+--R - log(tan(-------------))
+--R 2
+--R +
+--R log
+--R +-------+
+--R 2 2 2 | 2 2
+--R (p q sin(a x) - p cos(a x) - q - p )\|q + p
+--R +
+--R 3 2 2 3 2 3
+--R (- q - p q)sin(a x) + (p q + p )cos(a x) + p q + p
+--R /
+--R p sin(a x) + q cos(a x)
+--R /
+--R +-------+
+--R | 2 2
+--R a\|q + p
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.420~~~~~$\displaystyle
@@ -559,7 +1801,7 @@ $$
<<*>>=
)clear all
---S 27 of 36
+--S 141
aa:=integrate(1/(p*sin(a*x)+q*cos(a*x)+r),x)
--R
--R
@@ -600,6 +1842,174 @@ aa:=integrate(1/(p*sin(a*x)+q*cos(a*x)+r),x)
--R a\|r - q - p
--R Type: Union(List Expression
Integer,...)
--E
+
+--S 142
+bb1:=2/(a*sqrt(r^2-p^2-q^2))*atan((p+(r-q)*tan((a*x)/2))/sqrt(r^2-p^2-q^2))
+--R
+--R a x
+--R (r - q)tan(---) + p
+--R 2
+--R 2atan(-------------------)
+--R +------------+
+--R | 2 2 2
+--R \|r - q - p
+--R (2) --------------------------
+--R +------------+
+--R | 2 2 2
+--R a\|r - q - p
+--R Type: Expression
Integer
+--E
+
+--S 143
+bb2:=1/(a*sqrt(p^2+q^2-r^2))*log((p-sqrt(p^2+q^2-r^2)+(r-q)*tan((a*x)/2))/(p+sqrt(p^2+q^2-r^2)+(r-q)*tan((a*x)/2)))
+--R
+--R +--------------+
+--R | 2 2 2 a x
+--R - \|- r + q + p + (r - q)tan(---) + p
+--R 2
+--R log(-----------------------------------------)
+--R +--------------+
+--R | 2 2 2 a x
+--R \|- r + q + p + (r - q)tan(---) + p
+--R 2
+--R (3) ----------------------------------------------
+--R +--------------+
+--R | 2 2 2
+--R a\|- r + q + p
+--R Type: Expression
Integer
+--E
+
+--S 144
+cc1:=aa.1-bb1
+--R
+--R (4)
+--R +------------+
+--R | 2 2 2
+--R \|r - q - p
+--R *
+--R log
+--R 2 2
2
+--R (p r - p q)sin(a x) + (- r + q r + p )cos(a x) - q r +
q
+--R +
+--R 2
+--R p
+--R *
+--R +--------------+
+--R | 2 2 2
+--R \|- r + q + p
+--R +
+--R 3 2 2 2 3 2
+--R (r - q r + (- q - p )r + q + p q)sin(a x)
+--R +
+--R 2 2 3 2 2 3
+--R (p r - p q - p )cos(a x) + p r - p q - p
+--R /
+--R p sin(a x) + q cos(a x) + r
+--R +
+--R a x
+--R +--------------+ (r - q)tan(---) + p
+--R | 2 2 2 2
+--R - 2\|- r + q + p atan(-------------------)
+--R +------------+
+--R | 2 2 2
+--R \|r - q - p
+--R /
+--R +--------------+ +------------+
+--R | 2 2 2 | 2 2 2
+--R a\|- r + q + p \|r - q - p
+--R Type: Expression
Integer
+--E
+
+--S 145
+cc2:=aa.2-bb1
+--R
+--R (5)
+--R +------------+
+--R | 2 2 2
+--R ((r - q)sin(a x) + p cos(a x) + p)\|r - q - p
+--R 2atan(-------------------------------------------------)
+--R 2 2 2 2 2 2
+--R (r - q - p )cos(a x) + r - q - p
+--R +
+--R a x
+--R (r - q)tan(---) + p
+--R 2
+--R - 2atan(-------------------)
+--R +------------+
+--R | 2 2 2
+--R \|r - q - p
+--R /
+--R +------------+
+--R | 2 2 2
+--R a\|r - q - p
+--R Type: Expression
Integer
+--E
+
+--S 146
+cc3:=aa.1-bb2
+--R
+--R (6)
+--R log
+--R 2 2
2
+--R (p r - p q)sin(a x) + (- r + q r + p )cos(a x) - q r +
q
+--R +
+--R 2
+--R p
+--R *
+--R +--------------+
+--R | 2 2 2
+--R \|- r + q + p
+--R +
+--R 3 2 2 2 3 2
+--R (r - q r + (- q - p )r + q + p q)sin(a x)
+--R +
+--R 2 2 3 2 2 3
+--R (p r - p q - p )cos(a x) + p r - p q - p
+--R /
+--R p sin(a x) + q cos(a x) + r
+--R +
+--R +--------------+
+--R | 2 2 2 a x
+--R - \|- r + q + p + (r - q)tan(---) + p
+--R 2
+--R - log(-----------------------------------------)
+--R +--------------+
+--R | 2 2 2 a x
+--R \|- r + q + p + (r - q)tan(---) + p
+--R 2
+--R /
+--R +--------------+
+--R | 2 2 2
+--R a\|- r + q + p
+--R Type: Expression
Integer
+--E
+
+--S 147 14:420 Axiom cannot simplify these expressions
+cc4:=aa.2-bb2
+--R
+--R (7)
+--R +--------------+
+--R | 2 2 2 a x
+--R +------------+ - \|- r + q + p + (r - q)tan(---) + p
+--R | 2 2 2 2
+--R - \|r - q - p log(-----------------------------------------)
+--R +--------------+
+--R | 2 2 2 a x
+--R \|- r + q + p + (r - q)tan(---) + p
+--R 2
+--R +
+--R
+------------+
+--R +--------------+ | 2 2 2
+--R | 2 2 2 ((r - q)sin(a x) + p cos(a x) + p)\|r - q - p
+--R 2\|- r + q + p
atan(-------------------------------------------------)
+--R 2 2 2 2 2 2
+--R (r - q - p )cos(a x) + r - q - p
+--R /
+--R +--------------+ +------------+
+--R | 2 2 2 | 2 2 2
+--R a\|- r + q + p \|r - q - p
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.421~~~~~$\displaystyle
@@ -610,7 +2020,7 @@ $$
<<*>>=
)clear all
---S 28 of 36
+--S 148
aa:=integrate(1/(p*sin(a*x)+q*(1+cos(a*x))),x)
--R
--R
@@ -620,7 +2030,76 @@ aa:=integrate(1/(p*sin(a*x)+q*(1+cos(a*x))),x)
--R (1) --------------------------------
--R a p
--R Type: Union(Expression
Integer,...)
+--E
+
+--S 149
+bb:=1/(a*p)*log(q+p*tan((a*x)/2))
+--R
+--R a x
+--R log(p tan(---) + q)
+--R 2
+--R (2) -------------------
+--R a p
+--R Type: Expression
Integer
--E
+
+--S 150
+cc:=aa-bb
+--R
+--R a x p sin(a x) + q cos(a x) + q
+--R - log(p tan(---) + q) + log(---------------------------)
+--R 2 cos(a x) + 1
+--R (3) --------------------------------------------------------
+--R a p
+--R Type: Expression
Integer
+--E
+
+--S 151
+tanrule:=rule(tan(a) == sin(a)/cos(a))
+--R
+--R sin(a)
+--R (4) tan(a) == ------
+--R cos(a)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 152
+dd:=tanrule cc
+--R
+--R a x a x
+--R p sin(---) + q cos(---)
+--R p sin(a x) + q cos(a x) + q 2 2
+--R log(---------------------------) - log(-----------------------)
+--R cos(a x) + 1 a x
+--R cos(---)
+--R 2
+--R (5) ---------------------------------------------------------------
+--R a p
+--R Type: Expression
Integer
+--E
+
+--S 153
+ee:=expandLog dd
+--R
+--R (6)
+--R a x a x
+--R log(p sin(a x) + q cos(a x) + q) - log(p sin(---) + q cos(---))
+--R 2 2
+--R +
+--R a x
+--R - log(cos(a x) + 1) + log(cos(---))
+--R 2
+--R /
+--R a p
+--R Type: Expression
Integer
+--E
+
+--S 154 14:421 Schaums and Axiom agree
+ff:=complexNormalize ee
+--R
+--R (7) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.422~~~~~$\displaystyle
@@ -632,7 +2111,7 @@ $$
<<*>>=
)clear all
---S 29 of 36
+--S 155
aa:=integrate(1/(p*sin(a*x)+q*cos(a*x)+sqrt(p^2+q^2)),x)
--R
--R
@@ -665,9 +2144,88 @@ aa:=integrate(1/(p*sin(a*x)+q*cos(a*x)+sqrt(p^2+q^2)),x)
--R Type: Union(Expression
Integer,...)
--E
+--S 156
+bb:=-1/(a*sqrt(p^2+q^2))*tan(%pi/4-(a*x+atan(q/p))/2)
+--R
+--R q
+--R 2atan(-) + 2a x - %pi
+--R p
+--R tan(---------------------)
+--R 4
+--R (2) --------------------------
+--R +-------+
+--R | 2 2
+--R a\|q + p
+--R Type: Expression
Integer
+--E
+
+--S 157
+cc:=aa-bb
+--R
+--R (3)
+--R 6 2 4 4 2 6
+--R (64q + 80p q + 24p q + p )sin(a x)
+--R +
+--R 5 3 3 5 5 3 3 5
+--R (- 32p q - 32p q - 6p q)cos(a x) - 32p q - 32p q - 6p q
+--R *
+--R +-------+
+--R | 2 2
+--R \|q + p
+--R +
+--R 7 2 5 4 3 6
+--R (- 64q - 112p q - 56p q - 7p q)sin(a x)
+--R +
+--R 6 3 4 5 2 7 6 3 4 5 2
7
+--R (32p q + 48p q + 18p q + p )cos(a x) + 32p q + 48p q + 18p q
+ p
+--R *
+--R q
+--R 2atan(-) + 2a x - %pi
+--R p
+--R tan(---------------------)
+--R 4
+--R +
+--R 6 2 4 4 2 6 6 2 4 4 2
6
+--R ((64q + 96p q + 36p q + 2p )cos(a x) + 64q + 96p q + 36p q +
2p )
+--R *
+--R +-------+
+--R | 2 2
+--R \|q + p
+--R +
+--R 7 2 5 4 3 6 7 2 5 4 3
+--R (- 64q - 128p q - 76p q - 12p q)cos(a x) - 64q - 128p q - 76p q
+--R +
+--R 6
+--R - 12p q
+--R /
+--R 7 2 5 4 3 6
+--R (64a q + 112a p q + 56a p q + 7a p q)sin(a x)
+--R +
+--R 6 3 4 5 2 7 6
+--R (- 32a p q - 48a p q - 18a p q - a p )cos(a x) - 32a p q
+--R +
+--R 3 4 5 2 7
+--R - 48a p q - 18a p q - a p
+--R *
+--R +-------+
+--R | 2 2
+--R \|q + p
+--R +
+--R 8 2 6 4 4 6 2 8
+--R (- 64a q - 144a p q - 104a p q - 25a p q - a p )sin(a x)
+--R +
+--R 7 3 5 5 3 7 7
3 5
+--R (32a p q + 64a p q + 38a p q + 6a p q)cos(a x) + 32a p q + 64a p
q
+--R +
+--R 5 3 7
+--R 38a p q + 6a p q
+--R Type: Expression
Integer
+--E
+
+
)clear all
---S 30 of 36
+--S 158
aa:=integrate(1/(p*sin(a*x)+q*cos(a*x)-sqrt(p^2+q^2)),x)
--R
--R
@@ -699,6 +2257,84 @@ aa:=integrate(1/(p*sin(a*x)+q*cos(a*x)-sqrt(p^2+q^2)),x)
--R - 18a p q - a p
--R Type: Union(Expression
Integer,...)
--E
+
+--S 159
+bb:=-1/(a*sqrt(p^2+q^2))*tan(%pi/4+(a*x+atan(q/p))/2)
+--R
+--R q
+--R 2atan(-) + 2a x + %pi
+--R p
+--R tan(---------------------)
+--R 4
+--R (2) - --------------------------
+--R +-------+
+--R | 2 2
+--R a\|q + p
+--R Type: Expression
Integer
+--E
+
+--S 160 14:422 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R 6 2 4 4 2 6
+--R (64q + 80p q + 24p q + p )sin(a x)
+--R +
+--R 5 3 3 5 5 3 3 5
+--R (- 32p q - 32p q - 6p q)cos(a x) - 32p q - 32p q - 6p q
+--R *
+--R +-------+
+--R | 2 2
+--R \|q + p
+--R +
+--R 7 2 5 4 3 6
+--R (64q + 112p q + 56p q + 7p q)sin(a x)
+--R +
+--R 6 3 4 5 2 7 6 3 4
5 2
+--R (- 32p q - 48p q - 18p q - p )cos(a x) - 32p q - 48p q -
18p q
+--R +
+--R 7
+--R - p
+--R *
+--R q
+--R 2atan(-) + 2a x + %pi
+--R p
+--R tan(---------------------)
+--R 4
+--R +
+--R 6 2 4 4 2 6 6 2 4 4 2
6
+--R ((64q + 96p q + 36p q + 2p )cos(a x) + 64q + 96p q + 36p q +
2p )
+--R *
+--R +-------+
+--R | 2 2
+--R \|q + p
+--R +
+--R 7 2 5 4 3 6 7 2 5 4 3
6
+--R (64q + 128p q + 76p q + 12p q)cos(a x) + 64q + 128p q + 76p q +
12p q
+--R /
+--R 7 2 5 4 3 6
+--R (64a q + 112a p q + 56a p q + 7a p q)sin(a x)
+--R +
+--R 6 3 4 5 2 7 6
+--R (- 32a p q - 48a p q - 18a p q - a p )cos(a x) - 32a p q
+--R +
+--R 3 4 5 2 7
+--R - 48a p q - 18a p q - a p
+--R *
+--R +-------+
+--R | 2 2
+--R \|q + p
+--R +
+--R 8 2 6 4 4 6 2 8
+--R (64a q + 144a p q + 104a p q + 25a p q + a p )sin(a x)
+--R +
+--R 7 3 5 5 3 7 7
3 5
+--R (- 32a p q - 64a p q - 38a p q - 6a p q)cos(a x) - 32a p q - 64a
p q
+--R +
+--R 5 3 7
+--R - 38a p q - 6a p q
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.423~~~~~$\displaystyle
@@ -709,7 +2345,7 @@ $$
<<*>>=
)clear all
---S 31 of 36
+--S 161
aa:=integrate(1/(p^2*sin(a*x)^2+q^2*cos(a*x)^2),x)
--R
--R
@@ -722,6 +2358,37 @@ aa:=integrate(1/(p^2*sin(a*x)^2+q^2*cos(a*x)^2),x)
--R a p q
--R Type: Union(Expression
Integer,...)
--E
+
+--S 162
+bb:=1/(a*p*q)*atan((p*tan(a*x))/q)
+--R
+--R p tan(a x)
+--R atan(----------)
+--R q
+--R (2) ----------------
+--R a p q
+--R Type: Expression
Integer
+--E
+
+--S 163 14:423 Axiom cannot simplify this expressions
+cc:=aa-bb
+--R
+--R (3)
+--R 2 2 2
+--R p tan(a x) ((q - 2p )cos(a x) - 2p )sin(a x)
+--R - atan(----------) - atan(-----------------------------------)
+--R q 2
+--R p q cos(a x) + 2p q cos(a x) + p q
+--R +
+--R q sin(a x)
+--R atan(----------------)
+--R 2p cos(a x) + 2p
+--R /
+--R a p q
+--R Type: Expression
Integer
+--E
+
+
@
\section{\cite{1}:14.424~~~~~$\displaystyle
@@ -732,7 +2399,7 @@ $$
<<*>>=
)clear all
---S 32 of 36
+--S 164
aa:=integrate(1/(p^2*sin(a*x)^2-q^2*cos(a*x)^2),x)
--R
--R
@@ -742,7 +2409,77 @@ aa:=integrate(1/(p^2*sin(a*x)^2-q^2*cos(a*x)^2),x)
--R (1) -----------------------------------------------------------------
--R 2a p q
--R Type: Union(Expression
Integer,...)
+--E
+
+--S 165
+bb:=1/(2*a*p*q)*log((p*tan(a*x)-q)/(p*tan(a*x)+q))
+--R
+--R p tan(a x) - q
+--R log(--------------)
+--R p tan(a x) + q
+--R (2) -------------------
+--R 2a p q
+--R Type: Expression
Integer
--E
+
+--S 166
+cc:=aa-bb
+--R
+--R (3)
+--R 2p sin(a x) - 2q cos(a x) p tan(a x) - q
+--R log(-------------------------) - log(--------------)
+--R cos(a x) + 1 p tan(a x) + q
+--R +
+--R - 2p sin(a x) - 2q cos(a x)
+--R - log(---------------------------)
+--R cos(a x) + 1
+--R /
+--R 2a p q
+--R Type: Expression
Integer
+--E
+
+--S 167
+tanrule:=rule(tan(a) == sin(a)/cos(a))
+--R
+--R sin(a)
+--R (4) tan(a) == ------
+--R cos(a)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 168
+dd:=tanrule cc
+--R
+--R (5)
+--R 2p sin(a x) - 2q cos(a x) p sin(a x) - q cos(a x)
+--R log(-------------------------) - log(-----------------------)
+--R cos(a x) + 1 p sin(a x) + q cos(a x)
+--R +
+--R - 2p sin(a x) - 2q cos(a x)
+--R - log(---------------------------)
+--R cos(a x) + 1
+--R /
+--R 2a p q
+--R Type: Expression
Integer
+--E
+
+--S 169
+ee:=expandLog dd
+--R
+--R log(p sin(a x) + q cos(a x)) - log(- p sin(a x) - q cos(a x))
+--R (6) -------------------------------------------------------------
+--R 2a p q
+--R Type: Expression
Integer
+--E
+
+--S 170 14:424 Schaums and Axiom differ by a constant
+ff:=complexNormalize ee
+--R
+--R log(- 1)
+--R (7) --------
+--R 2a p q
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.425~~~~~$\displaystyle
@@ -763,7 +2500,7 @@ $$
<<*>>=
)clear all
---S 33 of 36
+--S 171 14:425 Axiom cannot compute this integral
aa:=integrate(sin(a*x)^m*cos(a*x)^n,x)
--R
--R
@@ -797,7 +2534,7 @@ $$
<<*>>=
)clear all
---S 34 of 36
+--S 172 14:426 Axiom cannot compute this integral
aa:=integrate(sin(a*x)^m/cos(a*x)^n,x)
--R
--R
@@ -832,7 +2569,7 @@ $$
<<*>>=
)clear all
---S 35 of 36
+--S 173 14:427 Axiom cannot compute this integral
aa:=integrate(cos(a*x)^m/sin(a*x)^n,x)
--R
--R
@@ -863,7 +2600,7 @@ $$
<<*>>=
)clear all
---S 36 of 36
+--S 174 14:428 Axiom cannot compute this integral
aa:=integrate(1/(sin(a*x)^m*cos(a*x)^n),x)
--R
--R
diff --git a/src/input/schaum20.input.pamphlet
b/src/input/schaum20.input.pamphlet
index a25209e..e6981a6 100644
--- a/src/input/schaum20.input.pamphlet
+++ b/src/input/schaum20.input.pamphlet
@@ -19,7 +19,7 @@ $$
)set message auto off
)clear all
---S 1 of 11
+--S 1
aa:=integrate(tan(a*x),x)
--R
--R
@@ -29,6 +29,81 @@ aa:=integrate(tan(a*x),x)
--R 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 2
+bb1:=-1/a*log(cos(a*x))
+--R
+--R log(cos(a x))
+--R (2) - -------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 3
+bb2:=1/a*log(sec(a*x))
+--R
+--R log(sec(a x))
+--R (3) -------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 4
+cc1:=aa-bb1
+--R
+--R 2
+--R log(tan(a x) + 1) + 2log(cos(a x))
+--R (4) -----------------------------------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 5
+tanrule:=rule(tan(a) == sin(a)/cos(a))
+--R
+--R sin(a)
+--R (5) tan(a) == ------
+--R cos(a)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 6
+dd1:=tanrule cc1
+--R
+--R 2 2
+--R sin(a x) + cos(a x)
+--R log(---------------------) + 2log(cos(a x))
+--R 2
+--R cos(a x)
+--R (6) -------------------------------------------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 7
+ee1:=expandLog dd1
+--R
+--R 2 2
+--R log(sin(a x) + cos(a x) )
+--R (7) --------------------------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 8
+sincossqrrule:=rule(sin(a)^2+cos(a)^2 == 1)
+--R
+--R 2 2
+--I (8) sin(a) + cos(a) + %K == %K + 1
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 9 14:429 Schaums and Axiom agree
+ff1:=sincossqrrule ee1
+--R
+--R (9) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.430~~~~~$\displaystyle
@@ -39,7 +114,7 @@ $$
<<*>>=
)clear all
---S 2 of 11
+--S 10
aa:=integrate(tan(a*x)^2,x)
--R
--R
@@ -48,6 +123,22 @@ aa:=integrate(tan(a*x)^2,x)
--R a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 11
+bb:=tan(a*x)/a-x
+--R
+--R tan(a x) - a x
+--R (2) --------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 12 14:430 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.431~~~~~$\displaystyle
@@ -58,7 +149,7 @@ $$
<<*>>=
)clear all
---S 3 of 11
+--S 13
aa:=integrate(tan(a*x)^3,x)
--R
--R
@@ -68,6 +159,73 @@ aa:=integrate(tan(a*x)^3,x)
--R 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 14
+bb:=tan(a*x)^2/(2*a)+1/a*log(cos(a*x))
+--R
+--R 2
+--R 2log(cos(a x)) + tan(a x)
+--R (2) --------------------------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 15
+cc:=aa-bb
+--R
+--R 2
+--R - log(tan(a x) + 1) - 2log(cos(a x))
+--R (3) -------------------------------------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 16
+tanrule:=rule(tan(a) == sin(a)/cos(a))
+--R
+--R sin(a)
+--R (4) tan(a) == ------
+--R cos(a)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 17
+dd:=tanrule cc
+--R
+--R 2 2
+--R sin(a x) + cos(a x)
+--R - log(---------------------) - 2log(cos(a x))
+--R 2
+--R cos(a x)
+--R (5) ---------------------------------------------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 18
+ee:=expandLog dd
+--R
+--R 2 2
+--R log(sin(a x) + cos(a x) )
+--R (6) - --------------------------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 19
+sincossqrrule:=rule(sin(a)^2+cos(a)^2 == 1)
+--R
+--R 2 2
+--I (7) sin(a) + cos(a) + %L == %L + 1
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 20 14:431 Schaums and Axiom agree
+ff:=sincossqrrule ee
+--R
+--R (8) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.432~~~~~$\displaystyle
@@ -78,7 +236,7 @@ $$
<<*>>=
)clear all
---S 4 of 11
+--S 21
aa:=integrate(tan(a*x)^n*sec(a*x)^2,x)
--R
--R
@@ -90,6 +248,74 @@ aa:=integrate(tan(a*x)^n*sec(a*x)^2,x)
--R (a n + a)cos(a x)
--R Type: Union(Expression
Integer,...)
--E
+
+--S 22
+bb:=tan(a*x)^(n+1)/((n+1)*a)
+--R
+--R n + 1
+--R tan(a x)
+--R (2) -------------
+--R a n + a
+--R Type: Expression
Integer
+--E
+
+--S 23
+cc:=aa-bb
+--R
+--R sin(a x)
+--R n log(--------)
+--R cos(a x) n + 1
+--R sin(a x)%e - cos(a x)tan(a x)
+--R (3) -------------------------------------------------
+--R (a n + a)cos(a x)
+--R Type: Expression
Integer
+--E
+
+--S 24
+explog:=rule(%e^(n*log(x)) == x^n)
+--R
+--R n log(x) n
+--R (4) %e == x
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 25
+dd:=explog cc
+--R
+--R n + 1 sin(a x) n
+--R - cos(a x)tan(a x) + sin(a x)(--------)
+--R cos(a x)
+--R (5) ---------------------------------------------
+--R (a n + a)cos(a x)
+--R Type: Expression
Integer
+--E
+
+--S 26
+tanrule:=rule(tan(a) == sin(a)/cos(a))
+--R
+--R sin(a)
+--R (6) tan(a) == ------
+--R cos(a)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 27
+ee:=tanrule dd
+--R
+--R sin(a x) n + 1 sin(a x) n
+--R - cos(a x)(--------) + sin(a x)(--------)
+--R cos(a x) cos(a x)
+--R (7) -----------------------------------------------
+--R (a n + a)cos(a x)
+--R Type: Expression
Integer
+--E
+
+--S 28 14:432 Schaums and Axiom agree
+ff:=complexNormalize ee
+--R
+--R (8) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.433~~~~~$\displaystyle
@@ -100,7 +326,7 @@ $$
<<*>>=
)clear all
---S 5 of 11
+--S 29
aa:=integrate(sec(a*x)^2/tan(a*x),x)
--R
--R
@@ -111,6 +337,44 @@ aa:=integrate(sec(a*x)^2/tan(a*x),x)
--R a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 30
+bb:=1/a*log(tan(a*x))
+--R
+--R log(tan(a x))
+--R (2) -------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 31
+cc:=aa-bb
+--R
+--R sin(a x) 2cos(a x)
+--R - log(tan(a x)) + log(------------) - log(- ------------)
+--R cos(a x) + 1 cos(a x) + 1
+--R (3) ---------------------------------------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 32
+dd:=expandLog cc
+--R
+--R - log(tan(a x)) + log(sin(a x)) - log(cos(a x)) - log(- 2)
+--R (4) ----------------------------------------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 33 14:433 Schaums and Axiom differ by a constant
+ee:=complexNormalize dd
+--R
+--R log(- 2)
+--R (5) - --------
+--R a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.434~~~~~$\displaystyle
@@ -121,7 +385,7 @@ $$
<<*>>=
)clear all
---S 6 of 11
+--S 34
aa:=integrate(1/tan(a*x),x)
--R
--R
@@ -131,6 +395,32 @@ aa:=integrate(1/tan(a*x),x)
--R 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 35
+bb:=1/a*log(sin(a*x))
+--R
+--R log(sin(a x))
+--R (2) -------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 36
+cc:=aa-bb
+--R
+--R 2
+--R - log(tan(a x) + 1) + 2log(tan(a x)) - 2log(sin(a x))
+--R (3) ------------------------------------------------------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 37
+complexNormalize cc
+--R
+--R (4) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.435~~~~~$\displaystyle
@@ -142,7 +432,7 @@ $$
<<*>>=
)clear all
---S 7 of 11
+--S 38 14:435 Axiom cannot compute this integral
aa:=integrate(x*tan(a*x),x)
--R
--R
@@ -163,7 +453,7 @@ $$
<<*>>=
)clear all
---S 8 of 11
+--S 39 14:436 Axiom cannot compute this integral
aa:=integrate(tan(a*x)/x,x)
--R
--R
@@ -183,7 +473,7 @@ $$
<<*>>=
)clear all
---S 9 of 11
+--S 40
aa:=integrate(x*tan(a*x)^2,x)
--R
--R
@@ -194,6 +484,77 @@ aa:=integrate(x*tan(a*x)^2,x)
--R 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 41
+bb:=(x*tan(a*x))/a+1/a^2*log(cos(a*x))-x^2/2
+--R
+--R 2 2
+--R 2log(cos(a x)) + 2a x tan(a x) - a x
+--R (2) -------------------------------------
+--R 2
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 42
+cc:=aa-bb
+--R
+--R 2
+--R - log(tan(a x) + 1) - 2log(cos(a x))
+--R (3) -------------------------------------
+--R 2
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 43
+tanrule:=rule(tan(a) == sin(a)/cos(a))
+--R
+--R sin(a)
+--R (4) tan(a) == ------
+--R cos(a)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 44
+dd:=tanrule cc
+--R
+--R 2 2
+--R sin(a x) + cos(a x)
+--R - log(---------------------) - 2log(cos(a x))
+--R 2
+--R cos(a x)
+--R (5) ---------------------------------------------
+--R 2
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 45
+ee:=expandLog dd
+--R
+--R 2 2
+--R log(sin(a x) + cos(a x) )
+--R (6) - --------------------------
+--R 2
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 46
+sincossqrrule:=rule(sin(a)^2+cos(a)^2 == 1)
+--R
+--R 2 2
+--I (7) sin(a) + cos(a) + %R == %R + 1
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 47 14:437 Schaums and Axiom agree
+ff:=sincossqrrule ee
+--R
+--R (8) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.438~~~~~$\displaystyle
@@ -204,7 +565,7 @@ $$
<<*>>=
)clear all
---S 10 of 11
+--S 48
aa:=integrate(1/(p+q*tan(a*x)),x)
--R
--R
@@ -215,6 +576,84 @@ aa:=integrate(1/(p+q*tan(a*x)),x)
--R 2a q + 2a p
--R Type: Union(Expression
Integer,...)
--E
+
+--S 49
+bb:=(p*x)/(p^2+q^2)+q/(a*(p^2+q^2))*log(q*sin(a*x)+p*cos(a*x))
+--R
+--R q log(q sin(a x) + p cos(a x)) + a p x
+--R (2) --------------------------------------
+--R 2 2
+--R a q + a p
+--R Type: Expression
Integer
+--E
+
+--S 50
+cc:=aa-bb
+--R
+--R (3)
+--R 2
+--R - q log(tan(a x) + 1) + 2q log(q tan(a x) + p)
+--R +
+--R - 2q log(q sin(a x) + p cos(a x))
+--R /
+--R 2 2
+--R 2a q + 2a p
+--R Type: Expression
Integer
+--E
+
+--S 51
+tanrule:=rule(tan(a) == sin(a)/cos(a))
+--R
+--R sin(a)
+--R (4) tan(a) == ------
+--R cos(a)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 52
+dd:=tanrule cc
+--R
+--R (5)
+--R 2 2
+--R sin(a x) + cos(a x)
+--R - q log(---------------------) - 2q log(q sin(a x) + p cos(a x))
+--R 2
+--R cos(a x)
+--R +
+--R q sin(a x) + p cos(a x)
+--R 2q log(-----------------------)
+--R cos(a x)
+--R /
+--R 2 2
+--R 2a q + 2a p
+--R Type: Expression
Integer
+--E
+
+--S 53
+ee:=expandLog dd
+--R
+--R 2 2
+--R q log(sin(a x) + cos(a x) )
+--R (6) - ----------------------------
+--R 2 2
+--R 2a q + 2a p
+--R Type: Expression
Integer
+--E
+
+--S 54
+sincossqrrule:=rule(sin(a)^2+cos(a)^2 == 1)
+--R
+--R 2 2
+--I (7) sin(a) + cos(a) + %S == %S + 1
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 55 14:438 Schaums and Axiom agree
+ff:=sincossqrrule ee
+--R
+--R (8) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.439~~~~~$\displaystyle
@@ -225,7 +664,7 @@ $$
<<*>>=
)clear all
---S 11 of 11
+--S 56 14:439 Axiom cannot compute this integral
aa:=integrate(tan(a*x)^n,x)
--R
--R
diff --git a/src/input/schaum21.input.pamphlet
b/src/input/schaum21.input.pamphlet
index 6e02ffe..4d6a8be 100644
--- a/src/input/schaum21.input.pamphlet
+++ b/src/input/schaum21.input.pamphlet
@@ -18,7 +18,7 @@ $$
)set message auto off
)clear all
---S 1 of 11
+--S 1
aa:=integrate(cot(a*x),x)
--R
--R
@@ -29,6 +29,42 @@ aa:=integrate(cot(a*x),x)
--R 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 2
+bb:=1/a*log(sin(a*x))
+--R
+--R log(sin(a x))
+--R (2) -------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 3
+cc:=aa-bb
+--R
+--R sin(2a x) 2
+--R 2log(-------------) - 2log(sin(a x)) - log(-------------)
+--R cos(2a x) + 1 cos(2a x) + 1
+--R (3) ---------------------------------------------------------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 4
+dd:=expandLog cc
+--R
+--R 2log(sin(2a x)) - 2log(sin(a x)) - log(cos(2a x) + 1) - log(2)
+--R (4) --------------------------------------------------------------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 5 14:440 Schaums and Axiom agree
+ee:=complexNormalize dd
+--R
+--R (5) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.441~~~~~$\displaystyle
@@ -39,7 +75,7 @@ $$
<<*>>=
)clear all
---S 2 of 11
+--S 6
aa:=integrate(cot(a*x)^2,x)
--R
--R
@@ -48,6 +84,49 @@ aa:=integrate(cot(a*x)^2,x)
--R a sin(2a x)
--R Type: Union(Expression
Integer,...)
--E
+
+--S 7
+bb:=-cot(a*x)/a-x
+--R
+--R - cot(a x) - a x
+--R (2) ----------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 8
+cc:=aa-bb
+--R
+--R cot(a x)sin(2a x) - cos(2a x) - 1
+--R (3) ---------------------------------
+--R a sin(2a x)
+--R Type: Expression
Integer
+--E
+
+--S 9
+cotrule:=rule(cot(a) == cos(a)/sin(a))
+--R
+--R cos(a)
+--R (4) cot(a) == ------
+--R sin(a)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 10
+dd:=cotrule cc
+--R
+--R cos(a x)sin(2a x) + (- cos(2a x) - 1)sin(a x)
+--R (5) ---------------------------------------------
+--R a sin(a x)sin(2a x)
+--R Type: Expression
Integer
+--E
+
+--S 11 14:441 Schaums and Axiom agree
+ee:=complexNormalize dd
+--R
+--R (6) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.442~~~~~$\displaystyle
@@ -58,7 +137,7 @@ $$
<<*>>=
)clear all
---S 3 of 11
+--S 12
aa:=integrate(cot(a*x)^3,x)
--R
--R
@@ -72,6 +151,97 @@ aa:=integrate(cot(a*x)^3,x)
--R 2a cos(2a x) - 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 13
+bb:=-cot(a*x)^2/(2*a)-1/a*log(sin(a*x))
+--R
+--R 2
+--R - 2log(sin(a x)) - cot(a x)
+--R (2) ----------------------------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 14
+cc:=aa-bb
+--R
+--R (3)
+--R sin(2a x)
+--R (- 2cos(2a x) + 2)log(-------------) + (2cos(2a x) - 2)log(sin(a x))
+--R cos(2a x) + 1
+--R +
+--R 2 2
+--R (cos(2a x) - 1)log(-------------) + (cos(2a x) - 1)cot(a x) +
cos(2a x)
+--R cos(2a x) + 1
+--R +
+--R 1
+--R /
+--R 2a cos(2a x) - 2a
+--R Type: Expression
Integer
+--E
+
+--S 15
+cotrule:=rule(cot(a) == cos(a)/sin(a))
+--R
+--R cos(a)
+--R (4) cot(a) == ------
+--R sin(a)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 16
+dd:=cotrule cc
+--R
+--R (5)
+--R 2 sin(2a x)
+--R (- 2cos(2a x) + 2)sin(a x) log(-------------)
+--R cos(2a x) + 1
+--R +
+--R 2
+--R (2cos(2a x) - 2)sin(a x) log(sin(a x))
+--R +
+--R 2 2 2
+--R (cos(2a x) - 1)sin(a x) log(-------------) + (cos(2a x) + 1)sin(a x)
+--R cos(2a x) + 1
+--R +
+--R 2 2
+--R cos(a x) cos(2a x) - cos(a x)
+--R /
+--R 2
+--R (2a cos(2a x) - 2a)sin(a x)
+--R Type: Expression
Integer
+--E
+
+--S 17
+ee:=expandLog dd
+--R
+--R (6)
+--R 2
+--R (- 2cos(2a x) + 2)sin(a x) log(sin(2a x))
+--R +
+--R 2
+--R (2cos(2a x) - 2)sin(a x) log(sin(a x))
+--R +
+--R 2
+--R (cos(2a x) - 1)sin(a x) log(cos(2a x) + 1)
+--R +
+--R 2 2
+--R ((log(2) + 1)cos(2a x) - log(2) + 1)sin(a x) + cos(a x) cos(2a x)
+--R +
+--R 2
+--R - cos(a x)
+--R /
+--R 2
+--R (2a cos(2a x) - 2a)sin(a x)
+--R Type: Expression
Integer
+--E
+
+--S 18 14:442 Schaums and Axiom agree
+ff:=complexNormalize ee
+--R
+--R (7) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.443~~~~~$\displaystyle
@@ -82,7 +252,7 @@ $$
<<*>>=
)clear all
---S 4 of 11
+--S 19
aa:=integrate(cot(a*x)^n*csc(a*x)^2,x)
--R
--R
@@ -94,6 +264,74 @@ aa:=integrate(cot(a*x)^n*csc(a*x)^2,x)
--R (a n + a)sin(a x)
--R Type: Union(Expression
Integer,...)
--E
+
+--S 20
+bb:=-cot(a*x)^(n+1)/((n+1)*a)
+--R
+--R n + 1
+--R cot(a x)
+--R (2) - -------------
+--R a n + a
+--R Type: Expression
Integer
+--E
+
+--S 21
+cc:=aa-bb
+--R
+--R cos(a x)
+--R n log(--------)
+--R sin(a x) n + 1
+--R - cos(a x)%e + sin(a x)cot(a x)
+--R (3) ---------------------------------------------------
+--R (a n + a)sin(a x)
+--R Type: Expression
Integer
+--E
+
+--S 22
+explog:=rule(%e^(n*log(x)) == x^n)
+--R
+--R n log(x) n
+--R (4) %e == x
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 23
+dd:=explog cc
+--R
+--R n + 1 cos(a x) n
+--R sin(a x)cot(a x) - cos(a x)(--------)
+--R sin(a x)
+--R (5) -------------------------------------------
+--R (a n + a)sin(a x)
+--R Type: Expression
Integer
+--E
+
+--S 24
+cotrule:=rule(cot(a) == cos(a)/sin(a))
+--R
+--R cos(a)
+--R (6) cot(a) == ------
+--R sin(a)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 25
+ee:=cotrule dd
+--R
+--R cos(a x) n + 1 cos(a x) n
+--R sin(a x)(--------) - cos(a x)(--------)
+--R sin(a x) sin(a x)
+--R (7) ---------------------------------------------
+--R (a n + a)sin(a x)
+--R Type: Expression
Integer
+--E
+
+--S 26 14:443 Schaums and Axiom agree
+ff:=complexNormalize ee
+--R
+--R (8) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.444~~~~~$\displaystyle
@@ -104,7 +342,7 @@ $$
<<*>>=
)clear all
---S 5 of 11
+--S 27
aa:=integrate(csc(a*x)^2/cot(a*x),x)
--R
--R
@@ -115,6 +353,55 @@ aa:=integrate(csc(a*x)^2/cot(a*x),x)
--R a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 28
+bb:=-1/a*log(cot(a*x))
+--R
+--R log(cot(a x))
+--R (2) - -------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 29
+cc:=aa-bb
+--R
+--R sin(a x) 2cos(a x)
+--R log(------------) + log(cot(a x)) - log(- ------------)
+--R cos(a x) + 1 cos(a x) + 1
+--R (3) -------------------------------------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 30
+cotrule:=rule(cot(a) == cos(a)/sin(a))
+--R
+--R cos(a)
+--R (4) cot(a) == ------
+--R sin(a)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 31
+dd:=cotrule cc
+--R
+--R sin(a x) cos(a x) 2cos(a x)
+--R log(------------) + log(--------) - log(- ------------)
+--R cos(a x) + 1 sin(a x) cos(a x) + 1
+--R (5) -------------------------------------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 32 14:444 Schaums and Axiom differ by a constant
+ee:=expandLog dd
+--R
+--R log(- 2)
+--R (6) - --------
+--R a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.445~~~~~$\displaystyle
@@ -125,7 +412,7 @@ $$
<<*>>=
)clear all
---S 6 of 11
+--S 33
aa:=integrate(1/cot(a*x),x)
--R
--R
@@ -136,6 +423,42 @@ aa:=integrate(1/cot(a*x),x)
--R 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 34
+bb:=-1/a*log(cos(a*x))
+--R
+--R log(cos(a x))
+--R (2) - -------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 35
+cc:=aa-bb
+--R
+--R 2
+--R 2log(cos(a x)) + log(-------------)
+--R cos(2a x) + 1
+--R (3) -----------------------------------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 36
+dd:=expandLog cc
+--R
+--R - log(cos(2a x) + 1) + 2log(cos(a x)) + log(2)
+--R (4) ----------------------------------------------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 37 14:445 Schaums and Axiom agree
+ee:=complexNormalize dd
+--R
+--R (5) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.446~~~~~$\displaystyle
@@ -148,7 +471,7 @@ $$
<<*>>=
)clear all
---S 7 of 11
+--S 38 14:446 Axiom cannot compute this integral
aa:=integrate(x*cot(a*x),x)
--R
--R
@@ -169,7 +492,7 @@ $$
<<*>>=
)clear all
---S 8 of 11
+--S 39 14:447 Axiom cannot compute this integral
aa:=integrate(cot(a*x)/x,x)
--R
--R
@@ -189,7 +512,7 @@ $$
<<*>>=
)clear all
---S 9 of 11
+--S 40
aa:=integrate(x*cot(a*x)^2,x)
--R
--R
@@ -205,6 +528,58 @@ aa:=integrate(x*cot(a*x)^2,x)
--R 2a sin(2a x)
--R Type: Union(Expression
Integer,...)
--E
+
+--S 41
+bb:=-(x*cot(a*x))/a+1/a^2*log(sin(a*x))-x^2/2
+--R
+--R 2 2
+--R 2log(sin(a x)) - 2a x cot(a x) - a x
+--R (2) -------------------------------------
+--R 2
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 42
+cc:=aa-bb
+--R
+--R (3)
+--R sin(2a x)
+--R 2sin(2a x)log(-------------) - 2sin(2a x)log(sin(a x))
+--R cos(2a x) + 1
+--R +
+--R 2
+--R - sin(2a x)log(-------------) + 2a x cot(a x)sin(2a x) - 2a x cos(2a
x)
+--R cos(2a x) + 1
+--R +
+--R - 2a x
+--R /
+--R 2
+--R 2a sin(2a x)
+--R Type: Expression
Integer
+--E
+
+--S 43
+dd:=expandLog cc
+--R
+--R (4)
+--R 2sin(2a x)log(sin(2a x)) - 2sin(2a x)log(sin(a x))
+--R +
+--R - sin(2a x)log(cos(2a x) + 1) + (2a x cot(a x) - log(2))sin(2a x)
+--R +
+--R - 2a x cos(2a x) - 2a x
+--R /
+--R 2
+--R 2a sin(2a x)
+--R Type: Expression
Integer
+--E
+
+--S 44 14:448 Schaums and Axiom agree
+ee:=complexNormalize dd
+--R
+--R (5) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.449~~~~~$\displaystyle
@@ -215,7 +590,7 @@ $$
<<*>>=
)clear all
---S 10 of 11
+--S 45
aa:=integrate(1/(p+q*cot(a*x)),x)
--R
--R
@@ -228,6 +603,34 @@ aa:=integrate(1/(p+q*cot(a*x)),x)
--R 2a q + 2a p
--R Type: Union(Expression
Integer,...)
--E
+
+--S 46
+bb:=(p*x)/(p^2+q^2)-q/(a*(p^2+q^2))*log(p*sin(a*x)+q*cos(a*x))
+--R
+--R - q log(p sin(a x) + q cos(a x)) + a p x
+--R (2) ----------------------------------------
+--R 2 2
+--R a q + a p
+--R Type: Expression
Integer
+--E
+
+--S 47 14:449 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R p sin(2a x) + q cos(2a x) + q
+--R - 2q log(-----------------------------) + 2q log(p sin(a x) + q
cos(a x))
+--R cos(2a x) + 1
+--R +
+--R 2
+--R q log(-------------)
+--R cos(2a x) + 1
+--R /
+--R 2 2
+--R 2a q + 2a p
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.450~~~~~$\displaystyle
@@ -238,7 +641,7 @@ $$
<<*>>=
)clear all
---S 11 of 11
+--S 48 14:450 Axiom cannot compute this integral
aa:=integrate(cot(a*x)^n,x)
--R
--R
diff --git a/src/input/schaum22.input.pamphlet
b/src/input/schaum22.input.pamphlet
index 66befb9..173338a 100644
--- a/src/input/schaum22.input.pamphlet
+++ b/src/input/schaum22.input.pamphlet
@@ -19,7 +19,7 @@ $$
)set message auto off
)clear all
---S 1 of 10
+--S 1
aa:=integrate(sec(a*x),x)
--R
--R
@@ -30,6 +30,172 @@ aa:=integrate(sec(a*x),x)
--R a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 2
+bb1:=1/a*log(sec(a*x)+tan(a*x))
+--R
+--R log(tan(a x) + sec(a x))
+--R (2) ------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 3
+bb2:=1/a*log(tan((a*x)/2+%pi/4))
+--R
+--R 2a x + %pi
+--R log(tan(----------))
+--R 4
+--R (3) --------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 4
+cc1:=aa-bb1
+--R
+--R (4)
+--R sin(a x) + cos(a x) + 1
+--R - log(tan(a x) + sec(a x)) + log(-----------------------)
+--R cos(a x) + 1
+--R +
+--R sin(a x) - cos(a x) - 1
+--R - log(-----------------------)
+--R cos(a x) + 1
+--R /
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 5
+tanrule:=rule(tan(a) == sin(a)/cos(a))
+--R
+--R sin(a)
+--R (5) tan(a) == ------
+--R cos(a)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 6
+dd1:=tanrule cc1
+--R
+--R (6)
+--R sin(a x) + cos(a x)sec(a x) sin(a x) + cos(a x) + 1
+--R - log(---------------------------) + log(-----------------------)
+--R cos(a x) cos(a x) + 1
+--R +
+--R sin(a x) - cos(a x) - 1
+--R - log(-----------------------)
+--R cos(a x) + 1
+--R /
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 7
+secrule:=rule(sec(a) == 1/cos(a))
+--R
+--R 1
+--R (7) sec(a) == ------
+--R cos(a)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 8
+ee1:=secrule dd1
+--R
+--R (8)
+--R sin(a x) + 1 sin(a x) + cos(a x) + 1
+--R - log(------------) + log(-----------------------)
+--R cos(a x) cos(a x) + 1
+--R +
+--R sin(a x) - cos(a x) - 1
+--R - log(-----------------------)
+--R cos(a x) + 1
+--R /
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 9
+ff1:=expandLog ee1
+--R
+--R (9)
+--R log(sin(a x) + cos(a x) + 1) - log(sin(a x) + 1)
+--R +
+--R - log(sin(a x) - cos(a x) - 1) + log(cos(a x))
+--R /
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 10
+gg1:=complexNormalize ff1
+--R
+--R log(- 1)
+--R (10) --------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 11
+cc2:=aa-bb2
+--R
+--R (11)
+--R 2a x + %pi sin(a x) + cos(a x) + 1
+--R - log(tan(----------)) + log(-----------------------)
+--R 4 cos(a x) + 1
+--R +
+--R sin(a x) - cos(a x) - 1
+--R - log(-----------------------)
+--R cos(a x) + 1
+--R /
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 12
+dd2:=tanrule cc2
+--R
+--R (12)
+--R sin(a x) + cos(a x) + 1 sin(a x) - cos(a x) - 1
+--R log(-----------------------) - log(-----------------------)
+--R cos(a x) + 1 cos(a x) + 1
+--R +
+--R 2a x + %pi
+--R sin(----------)
+--R 4
+--R - log(---------------)
+--R 2a x + %pi
+--R cos(----------)
+--R 4
+--R /
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 13
+ee2:=expandLog dd2
+--R
+--R (13)
+--R log(sin(a x) + cos(a x) + 1) - log(sin(a x) - cos(a x) - 1)
+--R +
+--R 2a x + %pi 2a x + %pi
+--R - log(sin(----------)) + log(cos(----------))
+--R 4 4
+--R /
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 14 14:451 Schaums and Axiom differ by a constant
+ff2:=complexNormalize ee2
+--R
+--R log(- 1)
+--R (14) --------
+--R a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.452~~~~~$\displaystyle
@@ -40,7 +206,7 @@ $$
<<*>>=
)clear all
---S 2 of 10
+--S 15
aa:=integrate(sec(a*x)^2,x)
--R
--R
@@ -49,6 +215,40 @@ aa:=integrate(sec(a*x)^2,x)
--R a cos(a x)
--R Type: Union(Expression
Integer,...)
--E
+
+--S 16
+bb:=tan(a*x)/a
+--R
+--R tan(a x)
+--R (2) --------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 17
+cc:=aa-bb
+--R
+--R - cos(a x)tan(a x) + sin(a x)
+--R (3) -----------------------------
+--R a cos(a x)
+--R Type: Expression
Integer
+--E
+
+--S 18
+tanrule:=rule(tan(a) == sin(a)/cos(a))
+--R
+--R sin(a)
+--R (4) tan(a) == ------
+--R cos(a)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 19 14:452 Schaums and Axiom agree
+dd:=tanrule cc
+--R
+--R (5) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.453~~~~~$\displaystyle
@@ -59,7 +259,7 @@ $$
<<*>>=
)clear all
---S 3 of 10
+--S 20
aa:=integrate(sec(a*x)^3,x)
--R
--R
@@ -76,6 +276,113 @@ aa:=integrate(sec(a*x)^3,x)
--R 2a cos(a x)
--R Type: Union(Expression
Integer,...)
--E
+
+--S 21
+bb:=(sec(a*x)*tan(a*x))/(2*a)+1/(2*a)*log(sec(a*x)+tan(a*x))
+--R
+--R log(tan(a x) + sec(a x)) + sec(a x)tan(a x)
+--R (2) -------------------------------------------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 22
+cc:=aa-bb
+--R
+--R (3)
+--R 2
+--R - cos(a x) log(tan(a x) + sec(a x))
+--R +
+--R 2 sin(a x) + cos(a x) + 1
+--R cos(a x) log(-----------------------)
+--R cos(a x) + 1
+--R +
+--R 2 sin(a x) - cos(a x) - 1 2
+--R - cos(a x) log(-----------------------) - cos(a x) sec(a x)tan(a x)
+--R cos(a x) + 1
+--R +
+--R sin(a x)
+--R /
+--R 2
+--R 2a cos(a x)
+--R Type: Expression
Integer
+--E
+
+--S 23
+tanrule:=rule(tan(a) == sin(a)/cos(a))
+--R
+--R sin(a)
+--R (4) tan(a) == ------
+--R cos(a)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 24
+dd:=tanrule cc
+--R
+--R (5)
+--R 2 sin(a x) + cos(a x)sec(a x)
+--R - cos(a x) log(---------------------------)
+--R cos(a x)
+--R +
+--R 2 sin(a x) + cos(a x) + 1
+--R cos(a x) log(-----------------------)
+--R cos(a x) + 1
+--R +
+--R 2 sin(a x) - cos(a x) - 1
+--R - cos(a x) log(-----------------------) + (- cos(a x)sec(a x) +
1)sin(a x)
+--R cos(a x) + 1
+--R /
+--R 2
+--R 2a cos(a x)
+--R Type: Expression
Integer
+--E
+
+--S 25
+secrule:=rule(sec(a) == 1/cos(a))
+--R
+--R 1
+--R (6) sec(a) == ------
+--R cos(a)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 26
+ee:=secrule dd
+--R
+--R (7)
+--R sin(a x) + 1 sin(a x) + cos(a x) + 1
+--R - log(------------) + log(-----------------------)
+--R cos(a x) cos(a x) + 1
+--R +
+--R sin(a x) - cos(a x) - 1
+--R - log(-----------------------)
+--R cos(a x) + 1
+--R /
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 27
+ff:=expandLog ee
+--R
+--R (8)
+--R log(sin(a x) + cos(a x) + 1) - log(sin(a x) + 1)
+--R +
+--R - log(sin(a x) - cos(a x) - 1) + log(cos(a x))
+--R /
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 28 14:453 Schaums and Axiom differ by a constant
+gg:=complexNormalize ff
+--R
+--R log(- 1)
+--R (9) --------
+--R 2a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.454~~~~~$\displaystyle
@@ -86,7 +393,7 @@ $$
<<*>>=
)clear all
---S 4 of 10
+--S 29
aa:=integrate(sec(a*x)^n*tan(a*x),x)
--R
--R 1
@@ -100,6 +407,31 @@ aa:=integrate(sec(a*x)^n*tan(a*x),x)
--R a n
--R Type: Union(Expression
Integer,...)
--E
+
+--S 30
+bb:=sec(a*x)^n/(n*a)
+--R
+--R n
+--R sec(a x)
+--R (2) ---------
+--R a n
+--R Type: Expression
Integer
+--E
+
+--S 31 14:454 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R 1
+--R n log(---------)
+--R 2
+--R cos(a x)
+--R ----------------
+--R 2 n
+--R %e - sec(a x)
+--R (3) ------------------------------
+--R a n
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.455~~~~~$\displaystyle
@@ -110,7 +442,7 @@ $$
<<*>>=
)clear all
---S 5 of 10
+--S 32
aa:=integrate(1/sec(a*x),x)
--R
--R
@@ -118,7 +450,23 @@ aa:=integrate(1/sec(a*x),x)
--R (1) --------
--R a
--R Type: Union(Expression
Integer,...)
+--E
+
+--S 33
+bb:=sin(a*x)/a
+--R
+--R sin(a x)
+--R (2) --------
+--R a
+--R Type: Expression
Integer
--E
+
+--S 34 14:455 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.456~~~~~$\displaystyle
@@ -130,7 +478,7 @@ $$
<<*>>=
)clear all
---S 6 of 10
+--S 35 14:456 Axiom cannot compute this integral
aa:=integrate(x*sec(a*x),x)
--R
--R
@@ -151,7 +499,7 @@ $$
<<*>>=
)clear all
---S 7 of 10
+--S 36 14:457 Axiom cannot compute this integral
aa:=integrate(sec(a*x)/x,x)
--R
--R
@@ -171,7 +519,7 @@ $$
<<*>>=
)clear all
---S 8 of 10
+--S 37
aa:=integrate(x*sec(a*x)^2,x)
--R
--R
@@ -184,6 +532,64 @@ aa:=integrate(x*sec(a*x)^2,x)
--R a cos(a x)
--R Type: Union(Expression
Integer,...)
--E
+
+--S 38
+bb:=x/a*tan(a*x)+1/a^2*log(cos(a*x))
+--R
+--R log(cos(a x)) + a x tan(a x)
+--R (2) ----------------------------
+--R 2
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 39
+cc:=aa-bb
+--R
+--R (3)
+--R 2
+--R - cos(a x)log(cos(a x)) - cos(a x)log(------------)
+--R cos(a x) + 1
+--R +
+--R 2cos(a x)
+--R cos(a x)log(- ------------) - a x cos(a x)tan(a x) + a x sin(a x)
+--R cos(a x) + 1
+--R /
+--R 2
+--R a cos(a x)
+--R Type: Expression
Integer
+--E
+
+--S 40
+tanrule:=rule(tan(a) == sin(a)/cos(a))
+--R
+--R sin(a)
+--R (4) tan(a) == ------
+--R cos(a)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 41
+dd:=tanrule cc
+--R
+--R 2 2cos(a x)
+--R - log(cos(a x)) - log(------------) + log(- ------------)
+--R cos(a x) + 1 cos(a x) + 1
+--R (5) ---------------------------------------------------------
+--R 2
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 42 14:458 Schaums and Axiom differ by a constant
+ee:=expandLog dd
+--R
+--R - log(2) + log(- 2)
+--R (6) -------------------
+--R 2
+--R a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.459~~~~~$\displaystyle
@@ -194,7 +600,7 @@ $$
<<*>>=
)clear all
---S 9 of 10
+--S 43
aa:=integrate(1/(q+p*sec(a*x)),x)
--R
--R
@@ -218,7 +624,136 @@ aa:=integrate(1/(q+p*sec(a*x)),x)
--R | 2 2
--R a q\|- q + p
--R Type: Union(List Expression
Integer,...)
+--E
+
+--S 44
+t1:=integrate(1/(p+q*cos(a*x)),x)
+--R
+--R (2)
+--R +-------+
+--R | 2 2 2 2
+--R (- p cos(a x) - q)\|q - p + (- q + p )sin(a x)
+--R log(--------------------------------------------------)
+--R q cos(a x) + p
+--R [-------------------------------------------------------,
+--R +-------+
+--R | 2 2
+--R a\|q - p
+--R +---------+
+--R | 2 2
+--R sin(a x)\|- q + p
+--R 2atan(-----------------------)
+--R (q + p)cos(a x) + q + p
+--R ------------------------------]
+--R +---------+
+--R | 2 2
+--R a\|- q + p
+--R Type: Union(List Expression
Integer,...)
--E
+
+--S 45
+bb1:=x/q-p/q*t1.1
+--R
+--R (3)
+--R +-------+
+--R | 2 2 2 2
+-------+
+--R (- p cos(a x) - q)\|q - p + (- q + p )sin(a x) | 2
2
+--R - p log(--------------------------------------------------) + a x\|q - p
+--R q cos(a x) + p
+--R
---------------------------------------------------------------------------
+--R +-------+
+--R | 2 2
+--R a q\|q - p
+--R Type: Expression
Integer
+--E
+
+--S 46
+bb2:=x/q-p/q*t1.2
+--R
+--R +---------+
+--R | 2 2 +---------+
+--R sin(a x)\|- q + p | 2 2
+--R - 2p atan(-----------------------) + a x\|- q + p
+--R (q + p)cos(a x) + q + p
+--R (4) ----------------------------------------------------
+--R +---------+
+--R | 2 2
+--R a q\|- q + p
+--R Type: Expression
Integer
+--E
+
+--S 47
+cc1:=aa.1-bb1
+--R
+--R (5)
+--R +-------+
+--R | 2 2 2 2
+--R (- p cos(a x) - q)\|q - p + (q - p )sin(a x)
+--R p log(------------------------------------------------)
+--R q cos(a x) + p
+--R +
+--R +-------+
+--R | 2 2 2 2
+--R (- p cos(a x) - q)\|q - p + (- q + p )sin(a x)
+--R p log(--------------------------------------------------)
+--R q cos(a x) + p
+--R /
+--R +-------+
+--R | 2 2
+--R a q\|q - p
+--R Type: Expression
Integer
+--E
+
+--S 48
+cc2:=aa.1-bb2
+--R
+--R (6)
+--R +-------+
+--R +---------+ | 2 2 2 2
+--R | 2 2 (- p cos(a x) - q)\|q - p + (q - p )sin(a x)
+--R p\|- q + p log(------------------------------------------------)
+--R q cos(a x) + p
+--R +
+--R +---------+
+--R +-------+ | 2 2
+--R | 2 2 sin(a x)\|- q + p
+--R 2p\|q - p atan(-----------------------)
+--R (q + p)cos(a x) + q + p
+--R /
+--R +---------+ +-------+
+--R | 2 2 | 2 2
+--R a q\|- q + p \|q - p
+--R Type: Expression
Integer
+--E
+
+--S 49
+cc3:=aa.2-bb1
+--R
+--R (7)
+--R +-------+
+--R +---------+ | 2 2 2 2
+--R | 2 2 (- p cos(a x) - q)\|q - p + (- q + p )sin(a x)
+--R p\|- q + p log(--------------------------------------------------)
+--R q cos(a x) + p
+--R +
+--R +---------+
+--R +-------+ | 2 2
+--R | 2 2 sin(a x)\|- q + p
+--R - 2p\|q - p atan(-----------------------)
+--R (q + p)cos(a x) + q + p
+--R /
+--R +---------+ +-------+
+--R | 2 2 | 2 2
+--R a q\|- q + p \|q - p
+--R Type: Expression
Integer
+--E
+
+--S 50 14:459 Schaums and Axiom agree
+cc4:=aa.2-bb2
+--R
+--R (8) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.460~~~~~$\displaystyle
@@ -230,7 +765,7 @@ $$
<<*>>=
)clear all
---S 10 of 10
+--S 51 14:460 Axiom cannot compute this integral
aa:=integrate(sec(a*x)^n,x)
--R
--R
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- [Axiom-developer] 20080423.01.tpd.patch (CATS Schaums-Axiom equivalence testing (17-22)),
daly <=