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[Axiom-developer] 20080426.01.tpd.patch (CATS Schaums-Axiom equivalence
|
From: |
daly |
|
Subject: |
[Axiom-developer] 20080426.01.tpd.patch (CATS Schaums-Axiom equivalence testing (26-28)) |
|
Date: |
Sun, 27 Apr 2008 01:07:54 -0500 |
On at least 2 of these problems Schaums and Axiom seem to disagree
on the results, namely:
14:569 SCHAUMS AND AXIOM DISAGREE?
14:571 SCHAUMS AND AXIOM DISAGREE?
as the compute result is not constant but does not seem to have a
simplification that eliminates x.
This needs to be studied further.
schaum26.input.pamphlet
14:525 Schaums and Axiom agree
14:526 Schaums and Axiom agree
14:527 Schaums and Axiom agree
14:528 Schaums and Axiom agree
14:529 Schaums and Axiom agree
14:530 Schaums and Axiom agree
14:531 Schaums and Axiom agree
14:532 Schaums and Axiom agree
14:533 Schaums and Axiom agree by definition
14:534 Axiom cannot compute this integral
14:535 Axiom cannot compute this integral
14:536 Axiom cannot compute this integral
14:537 Schaums and Axiom agree
14:538 Schaums and Axiom agree
14:539 Axiom cannot compute this integral
schaum27.input.pamphlet
14:540 Schaums and Axiom agree
14:541 Schaums and Axiom agree
14:542 Schaums and Axiom agree
14:543 Axiom cannot compute this integral
14:544 Axiom cannot compute this integral
14:545 Axiom cannot simplify this expression
14:546 Axiom cannot compute this integral
14:547 Schaums and Axiom agree
14:548 Axiom cannot simplify this expression
14:549 Schaums and Axiom differ by a constant
14:550 Axiom cannot simplify this expression
14:551 Schaums and Axiom agree
14:552 Schaums and Axiom agree
14:553 Axiom cannot simplify this expression
14:554 Axiom cannot simplify this expression
14:555 Axiom cannot simplify this expression
14:556 Axiom cannot simplify this expression
14:557 Axiom cannot compute this integral
14:558 Axiom cannot compute this integral
14:559 Axiom cannot compute this integral
14:560 Axiom cannot compute this integral
14:561 Axiom cannot compute this integral
schaum28.input.pamphlet
14:562 Schaums and Axiom agree
14:563 Schaums and Axiom agree
14:564 Schaums and Axiom agree
14:565 Axiom cannot compute this integral
14:566 Axiom cannot compute this integral
14:567 Schaums and Axiom agree
14:568 Axiom cannot compute this integral
14:569 SCHAUMS AND AXIOM DISAGREE?
14:570 Schaums and Axiom agree
14:571 SCHAUMS AND AXIOM DISAGREE?
14:572 Schaums and Axiom agree
14:573 Schaums and Axiom agree
14:574 Schaums and Axiom agree
14:575 Schaums and Axiom differ by a constant
14:576 Schaums and Axiom differ by a constant
14:577 Schaums and Axiom differ by a constant
14:578 Schaums and Axiom differ by a constant
14:579 Axiom cannot compute this integral
14:580 Axiom cannot compute this integral
14:581 Axiom cannot simplify this expression
14:582 Axiom cannot simplify this expression
14:583 Axiom cannot simplify this expression
14:584 Axiom cannot compute this integral
14:585 Axiom cannot compute this integral
14:586 Axiom cannot compute this integral
14:587 Axiom cannot compute this integral
14:588 Axiom cannot compute this integral
14:589 Axiom cannot compute this integral
========================================================================
diff --git a/changelog b/changelog
index 3956a7b..c94aa65 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,6 @@
+20080426 tpd src/input/schaum28.input show Schaums-Axiom equivalence
+20080426 tpd src/input/schaum27.input show Schaums-Axiom equivalence
+20080426 tpd src/input/schaum26.input show Schaums-Axiom equivalence
20080425 tpd src/input/schaum25.input show Schaums-Axiom equivalence
20080425 tpd src/input/schaum24.input show Schaums-Axiom equivalence
20080424 tpd src/input/schaum23.input show Schaums-Axiom equivalence
diff --git a/src/input/schaum26.input.pamphlet
b/src/input/schaum26.input.pamphlet
index 7bf7244..9bdeec5 100644
--- a/src/input/schaum26.input.pamphlet
+++ b/src/input/schaum26.input.pamphlet
@@ -18,13 +18,27 @@ $$
)set message auto off
)clear all
---S 1 of 16
+--S 1
aa:=integrate(log(x),x)
--R
--R
--R (1) x log(x) - x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 2
+bb:=x*log(x)-x
+--R
+--R (2) x log(x) - x
+--R Type: Expression
Integer
+--E
+
+--S 3 14:525 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.526~~~~~$\displaystyle
@@ -35,7 +49,7 @@ $$
<<*>>=
)clear all
---S 2 of 16
+--S 4
aa:=integrate(x*log(x),x)
--R
--R
@@ -44,7 +58,24 @@ aa:=integrate(x*log(x),x)
--R (1) --------------
--R 4
--R Type: Union(Expression
Integer,...)
+--E
+
+--S 5
+bb:=x^2/2*(log(x)-1/2)
+--R
+--R 2 2
+--R 2x log(x) - x
+--R (2) --------------
+--R 4
+--R Type: Expression
Integer
--E
+
+--S 6 14:526 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.527~~~~~$\displaystyle
@@ -55,7 +86,7 @@ $$
<<*>>=
)clear all
---S 3 of 16
+--S 7
aa:=integrate(x^m*log(x),x)
--R
--R
@@ -65,7 +96,55 @@ aa:=integrate(x^m*log(x),x)
--R 2
--R m + 2m + 1
--R Type: Union(Expression
Integer,...)
---E
+--E
+
+--S 8
+bb:=x^(m+1)/(m+1)*(log(x)-1/(m+1))
+--R
+--R m + 1
+--R ((m + 1)log(x) - 1)x
+--R (2) -------------------------
+--R 2
+--R m + 2m + 1
+--R Type: Expression
Integer
+--E
+
+--S 9
+cc:=aa-bb
+--R
+--R m log(x) m + 1
+--R ((m + 1)x log(x) - x)%e + ((- m - 1)log(x) + 1)x
+--R (3) -------------------------------------------------------------
+--R 2
+--R m + 2m + 1
+--R Type: Expression
Integer
+--E
+
+--S 10
+explog:=rule(%e^(n*log(x)) == x^n)
+--R
+--R n log(x) n
+--R (4) %e == x
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 11
+dd:=explog cc
+--R
+--R m + 1 m
+--R ((- m - 1)log(x) + 1)x + ((m + 1)x log(x) - x)x
+--R (5) -----------------------------------------------------
+--R 2
+--R m + 2m + 1
+--R Type: Expression
Integer
+--E
+
+--S 12 14:527 Schaums and Axiom agree
+ee:=complexNormalize dd
+--R
+--R (6) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.528~~~~~$\displaystyle
@@ -76,7 +155,7 @@ $$
<<*>>=
)clear all
---S 4 of 16
+--S 13
aa:=integrate(log(x)/x,x)
--R
--R
@@ -85,7 +164,24 @@ aa:=integrate(log(x)/x,x)
--R (1) -------
--R 2
--R Type: Union(Expression
Integer,...)
+--E
+
+--S 14
+bb:=1/2*log(x)^2
+--R
+--R 2
+--R log(x)
+--R (2) -------
+--R 2
+--R Type: Expression
Integer
--E
+
+--S 15 14:528 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.529~~~~~$\displaystyle
@@ -96,7 +192,7 @@ $$
<<*>>=
)clear all
---S 5 of 16
+--S 16
aa:=integrate(log(x)/x^2,x)
--R
--R
@@ -105,6 +201,22 @@ aa:=integrate(log(x)/x^2,x)
--R x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 17
+bb:=-log(x)/x-1/x
+--R
+--R - log(x) - 1
+--R (2) ------------
+--R x
+--R Type: Expression
Integer
+--E
+
+--S 18 14:529 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.530~~~~~$\displaystyle
@@ -115,14 +227,29 @@ $$
<<*>>=
)clear all
---S 6 of 16
+--S 19
aa:=integrate(log(x)^2,x)
--R
--R
--R 2
--R (1) x log(x) - 2x log(x) + 2x
--R Type: Union(Expression
Integer,...)
+--E
+
+--S 20
+bb:=x*log(x)^2-2*x*log(x)+2*x
+--R
+--R 2
+--R (2) x log(x) - 2x log(x) + 2x
+--R Type: Expression
Integer
--E
+
+--S 21 14:530 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.531~~~~~$\displaystyle
@@ -133,7 +260,7 @@ $$
<<*>>=
)clear all
---S 7 of 16
+--S 22
aa:=integrate(log(x)^n/x,x)
--R
--R
@@ -142,7 +269,52 @@ aa:=integrate(log(x)^n/x,x)
--R (1) ---------------------
--R n + 1
--R Type: Union(Expression
Integer,...)
+--E
+
+--S 23
+bb:=log(x)^(n+1)/(n+1)
+--R
+--R n + 1
+--R log(x)
+--R (2) -----------
+--R n + 1
+--R Type: Expression
Integer
--E
+
+--S 24
+cc:=aa-bb
+--R
+--R n log(log(x)) n + 1
+--R log(x)%e - log(x)
+--R (3) -----------------------------------
+--R n + 1
+--R Type: Expression
Integer
+--E
+
+--S 25
+explog:=rule(%e^(n*log(x)) == x^n)
+--R
+--R n log(x) n
+--R (4) %e == x
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 26
+dd:=explog cc
+--R
+--R n + 1 n
+--R - log(x) + log(x)log(x)
+--R (5) -----------------------------
+--R n + 1
+--R Type: Expression
Integer
+--E
+
+--S 27 14:531 Schaums and Axiom agree
+ee:=complexNormalize dd
+--R
+--R (6) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.532~~~~~$\displaystyle
@@ -153,13 +325,28 @@ $$
<<*>>=
)clear all
---S 8 of 16
+--S 28
aa:=integrate(1/(x*log(x)),x)
--R
--R
--R (1) log(log(x))
--R Type: Union(Expression
Integer,...)
--E
+
+--S 29
+bb:=log(log(x))
+--R
+--R (2) log(log(x))
+--R Type: Expression
Integer
+--E
+
+--S 30 14:532 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.533~~~~~$\displaystyle
@@ -171,7 +358,7 @@ $$
<<*>>=
)clear all
---S 9 of 16
+--S 31 14:533 Schaums and Axiom agree by definition
aa:=integrate(1/log(x),x)
--R
--R
@@ -189,7 +376,7 @@ $$
<<*>>=
)clear all
---S 10 of 16
+--S 32 14:534 Axiom cannot compute this integral
aa:=integrate(x^m/log(x),x)
--R
--R
@@ -209,7 +396,7 @@ $$
<<*>>=
)clear all
---S 11 of 16
+--S 33 14:535 Axiom cannot compute this integral
aa:=integrate(log(x)^n,x)
--R
--R
@@ -229,7 +416,7 @@ $$
<<*>>=
)clear all
---S 12 of 16
+--S 34 14:536 Axiom cannot compute this integral
aa:=integrate(x^m*log(x)^n,x)
--R
--R
@@ -249,7 +436,7 @@ $$
<<*>>=
)clear all
---S 13 of 16
+--S 35
aa:=integrate(log(x^2+a^2),x)
--R
--R
@@ -258,6 +445,22 @@ aa:=integrate(log(x^2+a^2),x)
--R a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 36
+bb:=x*log(x^2+a^2)-2*x+2*a*atan(x/a)
+--R
+--R 2 2 x
+--R (2) x log(x + a ) + 2a atan(-) - 2x
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 37 14:537 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.538~~~~~$\displaystyle
@@ -268,7 +471,7 @@ $$
<<*>>=
)clear all
---S 14 of 16
+--S 38
aa:=integrate(log(x^2-a^2),x)
--R
--R
@@ -276,6 +479,31 @@ aa:=integrate(log(x^2-a^2),x)
--R (1) x log(x - a ) + a log(x + a) - a log(x - a) - 2x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 39
+bb:=x*log(x^2-a^2)-2*x+a*log((x+a)/(x-a))
+--R
+--R 2 2 x + a
+--R (2) x log(x - a ) + a log(-----) - 2x
+--R x - a
+--R Type: Expression
Integer
+--E
+
+--S 40
+cc:=aa-bb
+--R
+--R x + a
+--R (3) a log(x + a) - a log(x - a) - a log(-----)
+--R x - a
+--R Type: Expression
Integer
+--E
+
+--S 41 14:538 Schaums and Axiom agree
+dd:=expandLog cc
+--R
+--R (4) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.539~~~~~$\displaystyle
@@ -287,7 +515,7 @@ $$
<<*>>=
)clear all
---S 15a of 16
+--S 42
aa:=integrate(x^m*log(x^2+a^2),x)
--R
--R
@@ -296,11 +524,11 @@ aa:=integrate(x^m*log(x^2+a^2),x)
--I (1) | log(a + %I )%I d%I
--R ++
--R Type: Union(Expression
Integer,...)
---E
+--E
)clear all
---S 15b of 16
+--S 43 14:539 Axiom cannot compute this integral
aa:=integrate(x^m*log(x^2-a^2),x)
--R
--R
diff --git a/src/input/schaum27.input.pamphlet
b/src/input/schaum27.input.pamphlet
index bb9f4d2..cc233e4 100644
--- a/src/input/schaum27.input.pamphlet
+++ b/src/input/schaum27.input.pamphlet
@@ -18,7 +18,7 @@ $$
)set message auto off
)clear all
---S 1 of 22
+--S 1
aa:=integrate(sinh(a*x),x)
--R
--R cosh(a x)
@@ -26,6 +26,22 @@ aa:=integrate(sinh(a*x),x)
--R a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 2
+bb:=cosh(a*x)/a
+--R
+--R cosh(a x)
+--R (2) ---------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 3 14:540 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.541~~~~~$\displaystyle
@@ -36,7 +52,7 @@ $$
<<*>>=
)clear all
---S 2 of 22
+--S 4
aa:=integrate(x*sinh(a*x),x)
--R
--R
@@ -46,6 +62,23 @@ aa:=integrate(x*sinh(a*x),x)
--R a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 5
+bb:=(x*cosh(a*x))/a-sinh(a*x)/a^2
+--R
+--R - sinh(a x) + a x cosh(a x)
+--R (2) ---------------------------
+--R 2
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 6 14:541 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.542~~~~~$\displaystyle
@@ -56,7 +89,7 @@ $$
<<*>>=
)clear all
---S 3 of 22
+--S 7
aa:=integrate(x^2*sinh(a*x),x)
--R
--R
@@ -67,6 +100,24 @@ aa:=integrate(x^2*sinh(a*x),x)
--R a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 8
+bb:=(x^2/a+2/a^3)*cosh(a*x)-(2*x)/a^2*sinh(a*x)
+--R
+--R 2 2
+--R - 2a x sinh(a x) + (a x + 2)cosh(a x)
+--R (2) --------------------------------------
+--R 3
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 9 14:542 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.543~~~~~$\displaystyle
@@ -77,7 +128,7 @@ $$
<<*>>=
)clear all
---S 4 of 22
+--S 10 14:543 Axiom cannot compute this integral
aa:=integrate(sinh(a*x)/x,x)
--R
--R
@@ -97,7 +148,7 @@ $$
<<*>>=
)clear all
---S 5 of 22
+--S 11 14:544 Axiom cannot compute this integral
aa:=integrate(sinh(a*x)/x^2,x)
--R
--R
@@ -118,7 +169,7 @@ $$
<<*>>=
)clear all
---S 6 of 22
+--S 12
aa:=integrate(1/sinh(a*x),x)
--R
--R
@@ -127,6 +178,31 @@ aa:=integrate(1/sinh(a*x),x)
--R a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 13
+bb:=1/a*log(tanh(a*x)/2)
+--R
+--R tanh(a x)
+--R log(---------)
+--R 2
+--R (2) --------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 14 14:545 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R tanh(a x)
+--R - log(---------) - log(sinh(a x) + cosh(a x) + 1)
+--R 2
+--R +
+--R log(sinh(a x) + cosh(a x) - 1)
+--R /
+--R a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.546~~~~~$\displaystyle
@@ -138,7 +214,7 @@ $$
<<*>>=
)clear all
---S 7 of 22
+--S 15 14:546 Axiom cannot compute this integral
aa:=integrate(x/sinh(a*x),x)
--R
--R
@@ -158,7 +234,7 @@ $$
<<*>>=
)clear all
---S 8 of 22
+--S 16
aa:=integrate(sinh(a*x)^2,x)
--R
--R
@@ -167,6 +243,22 @@ aa:=integrate(sinh(a*x)^2,x)
--R 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 17
+bb:=(sinh(a*x)*cosh(a*x))/(2*a)-x/2
+--R
+--R cosh(a x)sinh(a x) - a x
+--R (2) ------------------------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 18 14:547 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.548~~~~~$\displaystyle
@@ -177,7 +269,7 @@ $$
<<*>>=
)clear all
---S 9 of 22
+--S 19
aa:=integrate(x*sinh(a*x)^2,x)
--R
--R
@@ -188,6 +280,87 @@ aa:=integrate(x*sinh(a*x)^2,x)
--R 8a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 20
+bb:=(x*sinh(2*a*x))/(4*a)-cosh(2*a*x)/(8*a^2)-x^2/4
+--R
+--R 2 2
+--R 2a x sinh(2a x) - cosh(2a x) - 2a x
+--R (2) ------------------------------------
+--R 2
+--R 8a
+--R Type: Expression
Integer
+--E
+
+--S 21
+cc:=aa-bb
+--R
+--R (3)
+--R 2
+--R - 2a x sinh(2a x) - sinh(a x) + 4a x cosh(a x)sinh(a x) + cosh(2a x)
+--R +
+--R 2
+--R - cosh(a x)
+--R /
+--R 2
+--R 8a
+--R Type: Expression
Integer
+--E
+
+--S 22
+sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
+--R
+--R 2 cosh(2x) - 1
+--R (4) sinh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 23
+dd:=sinhsqrrule cc
+--R
+--R (5)
+--R 2
+--R - 4a x sinh(2a x) + 8a x cosh(a x)sinh(a x) + cosh(2a x) - 2cosh(a x) +
1
+--R
--------------------------------------------------------------------------
+--R 2
+--R 16a
+--R Type: Expression
Integer
+--E
+
+--S 24
+coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
+--R
+--R 2 cosh(2x) + 1
+--R (6) cosh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 25
+ee:=coshsqrrule dd
+--R
+--R - x sinh(2a x) + 2x cosh(a x)sinh(a x)
+--R (7) --------------------------------------
+--R 4a
+--R Type: Expression
Integer
+--E
+
+--S 26
+sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y)))
+--R
+--I %K sinh(y + x) - %K sinh(y - x)
+--I (8) %K cosh(y)sinh(x) == -------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 27 14:548 Axiom cannot simplify this expression
+ff:=sinhcoshrule ee
+--R
+--R (9) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.549~~~~~$\displaystyle
@@ -198,7 +371,7 @@ $$
<<*>>=
)clear all
---S 10 of 22
+--S 28
aa:=integrate(1/sinh(a*x)^2,x)
--R
--R
@@ -208,6 +381,158 @@ aa:=integrate(1/sinh(a*x)^2,x)
--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 29
+bb:=-coth(a*x)/a
+--R
+--R coth(a x)
+--R (2) - ---------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 30
+cc:=aa-bb
+--R
+--R (3)
+--R 2
+--R coth(a x)sinh(a x) + 2cosh(a x)coth(a x)sinh(a x)
+--R +
+--R 2
+--R (cosh(a x) - 1)coth(a x) - 2
+--R /
+--R 2 2
+--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a
+--R Type: Expression
Integer
+--E
+
+--S 31
+sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
+--R
+--R 2 cosh(2x) - 1
+--R (4) sinh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 32
+dd:=sinhsqrrule cc
+--R
+--R (5)
+--R 2
+--R 4cosh(a x)coth(a x)sinh(a x) + (cosh(2a x) + 2cosh(a x) - 3)coth(a x) -
4
+--R
--------------------------------------------------------------------------
+--R 2
+--R 4a cosh(a x)sinh(a x) + a cosh(2a x) + 2a cosh(a x) - 3a
+--R Type: Expression
Integer
+--E
+
+--S 33
+coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
+--R
+--R 2 cosh(2x) + 1
+--R (6) cosh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 34
+ee:=coshsqrrule dd
+--R
+--R 2cosh(a x)coth(a x)sinh(a x) + (cosh(2a x) - 1)coth(a x) - 2
+--R (7) ------------------------------------------------------------
+--R 2a cosh(a x)sinh(a x) + a cosh(2a x) - a
+--R Type: Expression
Integer
+--E
+
+--S 35
+sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y)))
+--I %B sinh(y + x) - %B sinh(y - x)
+--I (8) %B cosh(y)sinh(x) == -------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 36
+ff:=sinhcoshrule ee
+--R
+--R coth(a x)sinh(2a x) + (cosh(2a x) - 1)coth(a x) - 2
+--R (9) ---------------------------------------------------
+--R a sinh(2a x) + a cosh(2a x) - a
+--R Type: Expression
Integer
+--E
+
+--S 37
+cothrule:=rule(coth(x) == cosh(x)/sinh(x))
+--R
+--R cosh(x)
+--R (10) coth(x) == -------
+--R sinh(x)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 38
+gg:=cothrule ff
+--R
+--R cosh(a x)sinh(2a x) - 2sinh(a x) + cosh(a x)cosh(2a x) - cosh(a x)
+--R (11) ------------------------------------------------------------------
+--R a sinh(a x)sinh(2a x) + (a cosh(2a x) - a)sinh(a x)
+--R Type: Expression
Integer
+--E
+
+--S 39
+hh:=sinhcoshrule gg
+--R
+--R sinh(3a x) - 3sinh(a x) + 2cosh(a x)cosh(2a x) - 2cosh(a x)
+--R (12) -----------------------------------------------------------
+--R a sinh(3a x) + 2a sinh(a x)sinh(2a x) - 3a sinh(a x)
+--R Type: Expression
Integer
+--E
+
+--S 40
+sinhsinhrule:=rule(sinh(x)*sinh(y)==1/2*(cosh(x+y)-cosh(x-y)))
+--R
+--I %M cosh(y + x) - %M cosh(y - x)
+--I (13) %M sinh(x)sinh(y) == -------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 41
+ii:=sinhsinhrule gg
+--R
+--R 2cosh(a x)sinh(2a x) - 4sinh(a x) + 2cosh(a x)cosh(2a x) - 2cosh(a
x)
+--R (14)
---------------------------------------------------------------------
+--R (2a cosh(2a x) - 2a)sinh(a x) + a cosh(3a x) - a cosh(a x)
+--R Type: Expression
Integer
+--E
+
+--S 42
+coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y)))
+--R
+--I %N cosh(y + x) + %N cosh(y - x)
+--I (15) %N cosh(x)cosh(y) == -------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 43
+jj:=coshcoshrule ii
+--R
+--R 2cosh(a x)sinh(2a x) - 4sinh(a x) + cosh(3a x) - cosh(a x)
+--R (16) ----------------------------------------------------------
+--R (2a cosh(2a x) - 2a)sinh(a x) + a cosh(3a x) - a cosh(a x)
+--R Type: Expression
Integer
+--E
+
+--S 44 14:549 Schaums and Axiom differ by a constant
+kk:=sinhcoshrule jj
+--R
+--R 1
+--R (17) -
+--R a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.550~~~~~$\displaystyle
@@ -218,7 +543,7 @@ $$
<<*>>=
)clear all
---S 11 of 22
+--S 45
aa:=integrate(sinh(a*x)*sinh(p*x),x)
--R
--R
@@ -228,6 +553,36 @@ aa:=integrate(sinh(a*x)*sinh(p*x),x)
--R (p - a )sinh(a x) + (- p + a )cosh(a x)
--R Type: Union(Expression
Integer,...)
--E
+
+--S 46
+bb:=(sinh(a+p)*x)/(2*(a+p))-(sinh(a-p)*x)/(2*(a-p))
+--R
+--R (p - a)x sinh(p + a) + (- p - a)x sinh(p - a)
+--R (2) ---------------------------------------------
+--R 2 2
+--R 2p - 2a
+--R Type: Expression
Integer
+--E
+
+--S 47 14:550 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R 2a cosh(a x)sinh(p x)
+--R +
+--R 2
+--R ((- p + a)x sinh(p + a) + (p + a)x sinh(p - a))sinh(a x)
+--R +
+--R 2
+--R - 2p cosh(p x)sinh(a x) + (p - a)x cosh(a x) sinh(p + a)
+--R +
+--R 2
+--R (- p - a)x cosh(a x) sinh(p - a)
+--R /
+--R 2 2 2 2 2 2
+--R (2p - 2a )sinh(a x) + (- 2p + 2a )cosh(a x)
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.551~~~~~$\displaystyle
@@ -238,7 +593,7 @@ $$
<<*>>=
)clear all
---S 12 of 22
+--S 48
aa:=integrate(sinh(a*x)*sin(p*x),x)
--R
--R
@@ -255,6 +610,71 @@ aa:=integrate(sinh(a*x)*sin(p*x),x)
--R (2p + 2a )sinh(a x) + (2p + 2a )cosh(a x)
--R Type: Union(Expression
Integer,...)
--E
+
+--S 49
+bb:=(a*cosh(a*x)*sin(p*x)-p*sinh(a*x)*cos(p*x))/(a^2+p^2)
+--R
+--R - p cos(p x)sinh(a x) + a cosh(a x)sin(p x)
+--R (2) -------------------------------------------
+--R 2 2
+--R p + a
+--R Type: Expression
Integer
+--E
+
+--S 50
+cc:=aa-bb
+--R
+--R (3)
+--R 2 2
+--R (a sin(p x) + p cos(p x))sinh(a x) + (- a cosh(a x) + a)sin(p x)
+--R +
+--R 2
+--R - p cos(p x)cosh(a x) + p cos(p x)
+--R /
+--R 2 2 2 2
+--R (2p + 2a )sinh(a x) + (2p + 2a )cosh(a x)
+--R Type: Expression
Integer
+--E
+
+--S 51
+sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
+--R
+--R 2 cosh(2x) - 1
+--R (4) sinh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 52
+dd:=sinhsqrrule cc
+--R
+--R (5)
+--R 2
+--R (a cosh(2a x) - 2a cosh(a x) + a)sin(p x) + p cos(p x)cosh(2a x)
+--R +
+--R 2
+--R - 2p cos(p x)cosh(a x) + p cos(p x)
+--R /
+--R 2 2 2 2
+--R (4p + 4a )sinh(a x) + (4p + 4a )cosh(a x)
+--R Type: Expression
Integer
+--E
+
+--S 53
+coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
+--R
+--R 2 cosh(2x) + 1
+--R (6) cosh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 54 14:551 Schaums and Axiom agree
+ee:=coshsqrrule dd
+--R
+--R (7) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.552~~~~~$\displaystyle
@@ -265,7 +685,7 @@ $$
<<*>>=
)clear all
---S 13 of 22
+--S 55
aa:=integrate(sinh(a*x)*cos(p*x),x)
--R
--R
@@ -282,6 +702,71 @@ aa:=integrate(sinh(a*x)*cos(p*x),x)
--R (2p + 2a )sinh(a x) + (2p + 2a )cosh(a x)
--R Type: Union(Expression
Integer,...)
--E
+
+--S 56
+bb:=(a*cosh(a*x)*cos(p*x)+p*sinh(a*x)*sin(p*x))/(a^2+p^2)
+--R
+--R p sin(p x)sinh(a x) + a cos(p x)cosh(a x)
+--R (2) -----------------------------------------
+--R 2 2
+--R p + a
+--R Type: Expression
Integer
+--E
+
+--S 57
+cc:=aa-bb
+--R
+--R (3)
+--R 2 2
+--R (- p sin(p x) + a cos(p x))sinh(a x) + (p cosh(a x) - p)sin(p x)
+--R +
+--R 2
+--R - a cos(p x)cosh(a x) + a cos(p x)
+--R /
+--R 2 2 2 2
+--R (2p + 2a )sinh(a x) + (2p + 2a )cosh(a x)
+--R Type: Expression
Integer
+--E
+
+--S 58
+sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
+--R
+--R 2 cosh(2x) - 1
+--R (4) sinh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 59
+dd:=sinhsqrrule cc
+--R
+--R (5)
+--R 2
+--R (- p cosh(2a x) + 2p cosh(a x) - p)sin(p x) + a cos(p x)cosh(2a x)
+--R +
+--R 2
+--R - 2a cos(p x)cosh(a x) + a cos(p x)
+--R /
+--R 2 2 2 2
+--R (4p + 4a )sinh(a x) + (4p + 4a )cosh(a x)
+--R Type: Expression
Integer
+--E
+
+--S 60
+coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
+--R
+--R 2 cosh(2x) + 1
+--R (6) cosh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 61 14:552 Schaums and Axiom agree
+ee:=coshsqrrule dd
+--R
+--R (7) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.553~~~~~$\displaystyle
@@ -293,7 +778,7 @@ $$
<<*>>=
)clear all
---S 14 of 22
+--S 62
aa:=integrate(1/(p+q*sinh(a*x)),x)
--R
--R
@@ -322,6 +807,60 @@ aa:=integrate(1/(p+q*sinh(a*x)),x)
--R a\|q + p
--R Type: Union(Expression
Integer,...)
--E
+
+--S 63
+bb:=1/(a*sqrt(p^2+q^2))*log((q*%e^(a*x)+p-sqrt(p^2+q^2))/(q*%e^(a*x)+p+sqrt(p^2+q^2)))
+--R
+--R +-------+
+--R | 2 2 a x
+--R - \|q + p + q %e + p
+--R log(--------------------------)
+--R +-------+
+--R | 2 2 a x
+--R \|q + p + q %e + p
+--R (2) -------------------------------
+--R +-------+
+--R | 2 2
+--R a\|q + p
+--R Type: Expression
Integer
+--E
+
+--S 64 14:553 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R log
+--R 2 2 2 2
2
+--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a
x)
+--R +
+--R 2 2
+--R 2p q cosh(a x) + q + 2p
+--R *
+--R +-------+
+--R | 2 2
+--R \|q + p
+--R +
+--R 3 2 3 2 2
3
+--R (- 2q - 2p q)sinh(a x) + (- 2q - 2p q)cosh(a x) - 2p q - 2p
+--R /
+--R 2 2
+--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x)
+--R +
+--R 2p cosh(a x) - q
+--R +
+--R +-------+
+--R | 2 2 a x
+--R - \|q + p + q %e + p
+--R - log(--------------------------)
+--R +-------+
+--R | 2 2 a x
+--R \|q + p + q %e + p
+--R /
+--R +-------+
+--R | 2 2
+--R a\|q + p
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.554~~~~~$\displaystyle
@@ -333,7 +872,7 @@ $$
<<*>>=
)clear all
---S 15 of 22
+--S 65
aa:=integrate(1/(p*q*sinh(a*x))^2,x)
--R
--R
@@ -344,6 +883,137 @@ aa:=integrate(1/(p*q*sinh(a*x))^2,x)
--R a p q sinh(a x) + 2a p q cosh(a x)sinh(a x) + a p q cosh(a x) - a p q
--R Type: Union(Expression
Integer,...)
--E
+
+--S 66
+t1:=integrate(1/(p+q*sinh(a*x)),x)
+--R
+--R (2)
+--R log
+--R 2 2 2 2 2
+--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x)
+--R +
+--R 2 2
+--R 2p q cosh(a x) + q + 2p
+--R *
+--R +-------+
+--R | 2 2
+--R \|q + p
+--R +
+--R 3 2 3 2 2 3
+--R (- 2q - 2p q)sinh(a x) + (- 2q - 2p q)cosh(a x) - 2p q - 2p
+--R /
+--R 2 2
+--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x)
+--R +
+--R 2p cosh(a x) - q
+--R /
+--R +-------+
+--R | 2 2
+--R a\|q + p
+--R Type: Union(Expression
Integer,...)
+--E
+
+--S 67
+bb:=(-q*cosh(a*x))/(a*(p^2+q^2)*(p+q*sinh(a*x)))+p/(p^2+q^2)*t1
+--R
+--R (3)
+--R 2
+--R (p q sinh(a x) + p )
+--R *
+--R log
+--R 2 2 2 2
2
+--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q
cosh(a x)
+--R +
+--R 2 2
+--R 2p q cosh(a x) + q + 2p
+--R *
+--R +-------+
+--R | 2 2
+--R \|q + p
+--R +
+--R 3 2 3 2 2
3
+--R (- 2q - 2p q)sinh(a x) + (- 2q - 2p q)cosh(a x) - 2p q -
2p
+--R /
+--R 2 2
+--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x)
+--R +
+--R 2p cosh(a x) - q
+--R +
+--R +-------+
+--R | 2 2
+--R - q cosh(a x)\|q + p
+--R /
+--R +-------+
+--R 3 2 2 3 | 2 2
+--R ((a q + a p q)sinh(a x) + a p q + a p )\|q + p
+--R Type: Expression
Integer
+--E
+
+--S 68 14:554 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (4)
+--R 3 3 3 3 3 4 2 2
+--R - p q sinh(a x) + (- 2p q cosh(a x) - p q )sinh(a x)
+--R +
+--R 3 3 2 4 2 3 3 4 2
2
+--R (- p q cosh(a x) - 2p q cosh(a x) + p q )sinh(a x) - p q cosh(a
x)
+--R +
+--R 4 2
+--R p q
+--R *
+--R log
+--R 2 2 2 2
2
+--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q
cosh(a x)
+--R +
+--R 2 2
+--R 2p q cosh(a x) + q + 2p
+--R *
+--R +-------+
+--R | 2 2
+--R \|q + p
+--R +
+--R 3 2 3 2 2
3
+--R (- 2q - 2p q)sinh(a x) + (- 2q - 2p q)cosh(a x) - 2p q -
2p
+--R /
+--R 2 2
+--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x)
+--R +
+--R 2p cosh(a x) - q
+--R +
+--R 2 3 2 2 3 2 3 2
+--R p q cosh(a x)sinh(a x) + (2p q cosh(a x) - 2q - 2p q)sinh(a x)
+--R +
+--R 2 3 3 2 3 2 3
+--R p q cosh(a x) - p q cosh(a x) - 2p q - 2p
+--R *
+--R +-------+
+--R | 2 2
+--R \|q + p
+--R /
+--R 2 5 4 3 3
+--R (a p q + a p q )sinh(a x)
+--R +
+--R 2 5 4 3 3 4 5 2 2
+--R ((2a p q + 2a p q )cosh(a x) + a p q + a p q )sinh(a x)
+--R +
+--R 2 5 4 3 2 3 4 5 2
2 5
+--R (a p q + a p q )cosh(a x) + (2a p q + 2a p q )cosh(a x) - a
p q
+--R +
+--R 4 3
+--R - a p q
+--R *
+--R sinh(a x)
+--R +
+--R 3 4 5 2 2 3 4 5 2
+--R (a p q + a p q )cosh(a x) - a p q - a p q
+--R *
+--R +-------+
+--R | 2 2
+--R \|q + p
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.555~~~~~$\displaystyle
@@ -363,7 +1033,7 @@ $$
<<*>>=
)clear all
---S 16 of 22
+--S 69
aa:=integrate(1/(p^2+q^2*sinh(a*x)^2),x)
--R
--R
@@ -426,6 +1096,180 @@ aa:=integrate(1/(p^2+q^2*sinh(a*x)^2),x)
--R ]
--R Type: Union(List Expression
Integer,...)
--E
+
+--S 70
+bb1:=1/(a*p*sqrt(q^2-p^2))*atan((sqrt(q^2-p^2)*tanh(a*x))/p)
+--R
+--R +-------+
+--R | 2 2
+--R tanh(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 71
+bb2:=1/(2*a*p*sqrt(p^2-q^2))*log((p+sqrt(p^2-q^2)*tanh(a*x))/(p-sqrt(p^2-q^2)*tanh(a*x)))
+--R
+--R +---------+
+--R | 2 2
+--R - tanh(a x)\|- q + p - p
+--R log(---------------------------)
+--R +---------+
+--R | 2 2
+--R tanh(a x)\|- q + p - p
+--R (3) --------------------------------
+--R +---------+
+--R | 2 2
+--R 2a p\|- q + p
+--R Type: Expression
Integer
+--E
+
+--S 72
+cc1:=aa.1-bb1
+--R
+--R (4)
+--R +-------+
+--R | 2 2
+--R \|q - p
+--R *
+--R log
+--R 4 4 4 3
+--R q sinh(a x) + 4q cosh(a x)sinh(a x)
+--R +
+--R 4 2 4 2 2 2
+--R (6q cosh(a x) - 2q + 4p q )sinh(a x)
+--R +
+--R 4 3 4 2 2
+--R (4q cosh(a x) + (- 4q + 8p q )cosh(a x))sinh(a x)
+--R +
+--R 4 4 4 2 2 2 4 2 2
4
+--R q cosh(a x) + (- 2q + 4p q )cosh(a x) + q - 8p q +
8p
+--R *
+--R +---------+
+--R | 2 2
+--R \|- q + p
+--R +
+--R 4 3 2 2 4 3 2
+--R (4p q - 4p q )sinh(a x) + (8p q - 8p q )cosh(a x)sinh(a
x)
+--R +
+--R 4 3 2 2 4 3 2 5
+--R (4p q - 4p q )cosh(a x) - 4p q + 12p q - 8p
+--R /
+--R 2 4 2 3
+--R q sinh(a x) + 4q cosh(a x)sinh(a x)
+--R +
+--R 2 2 2 2 2
+--R (6q cosh(a x) - 2q + 4p )sinh(a x)
+--R +
+--R 2 3 2 2 2
4
+--R (4q cosh(a x) + (- 4q + 8p )cosh(a x))sinh(a x) + q
cosh(a x)
+--R +
+--R 2 2 2 2
+--R (- 2q + 4p )cosh(a x) + q
+--R +
+--R +-------+
+--R +---------+ | 2 2
+--R | 2 2 tanh(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 73
+cc2:=aa.2-bb1
+--R
+--R (5)
+--R +-------+
+--R | 2 2
+--R tanh(a x)\|q - p
+--R - atan(-------------------)
+--R p
+--R +
+--R atan
+--R 2 2 2 2 2 2
2
+--R (q sinh(a x) + 2q cosh(a x)sinh(a x) + q cosh(a x) - q +
2p )
+--R *
+--R +-------+
+--R | 2 2
+--R \|q - p
+--R /
+--R 2 3
+--R 2p q - 2p
+--R /
+--R +-------+
+--R | 2 2
+--R a p\|q - p
+--R Type: Expression
Integer
+--E
+
+--S 74
+cc3:=aa.2-bb1
+--R
+--R (6)
+--R +-------+
+--R | 2 2
+--R tanh(a x)\|q - p
+--R - atan(-------------------)
+--R p
+--R +
+--R atan
+--R 2 2 2 2 2 2
2
+--R (q sinh(a x) + 2q cosh(a x)sinh(a x) + q cosh(a x) - q +
2p )
+--R *
+--R +-------+
+--R | 2 2
+--R \|q - p
+--R /
+--R 2 3
+--R 2p q - 2p
+--R /
+--R +-------+
+--R | 2 2
+--R a p\|q - p
+--R Type: Expression
Integer
+--E
+
+--S 75 14:555 Axiom cannot simplify this expression
+cc4:=aa.2-bb2
+--R
+--R (7)
+--R +---------+
+--R +-------+ | 2 2
+--R | 2 2 - tanh(a x)\|- q + p - p
+--R - \|q - p log(---------------------------)
+--R +---------+
+--R | 2 2
+--R tanh(a x)\|- q + p - p
+--R +
+--R +---------+
+--R | 2 2
+--R 2\|- q + p
+--R *
+--R atan
+--R 2 2 2 2 2 2
2
+--R (q sinh(a x) + 2q cosh(a x)sinh(a x) + q cosh(a x) - q +
2p )
+--R *
+--R +-------+
+--R | 2 2
+--R \|q - p
+--R /
+--R 2 3
+--R 2p q - 2p
+--R /
+--R +---------+ +-------+
+--R | 2 2 | 2 2
+--R 2a p\|- q + p \|q - p
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.556~~~~~$\displaystyle
@@ -437,7 +1281,7 @@ $$
<<*>>=
)clear all
---S 17 of 22
+--S 76
aa:=integrate(1/(p^2+q^2*sinh(a*x)^2),x)
--R
--R
@@ -500,6 +1344,113 @@ aa:=integrate(1/(p^2+q^2*sinh(a*x)^2),x)
--R ]
--R Type: Union(List Expression
Integer,...)
--E
+
+--S 77
+bb:=1/(2*a*p*sqrt(p^2+q^2))*log((p+sqrt(p^2+q^2)*tanh(a*x))/(p-sqrt(p^2+q^2)*tanh(a*x)))
+--R
+--R +-------+
+--R | 2 2
+--R - tanh(a x)\|q + p - p
+--R log(-------------------------)
+--R +-------+
+--R | 2 2
+--R tanh(a x)\|q + p - p
+--R (2) ------------------------------
+--R +-------+
+--R | 2 2
+--R 2a p\|q + p
+--R Type: Expression
Integer
+--E
+
+--S 78
+cc1:=aa.1-bb
+--R
+--R (3)
+--R +-------+
+--R | 2 2
+--R \|q + p
+--R *
+--R log
+--R 4 4 4 3
+--R q sinh(a x) + 4q cosh(a x)sinh(a x)
+--R +
+--R 4 2 4 2 2 2
+--R (6q cosh(a x) - 2q + 4p q )sinh(a x)
+--R +
+--R 4 3 4 2 2
+--R (4q cosh(a x) + (- 4q + 8p q )cosh(a x))sinh(a x)
+--R +
+--R 4 4 4 2 2 2 4 2 2
4
+--R q cosh(a x) + (- 2q + 4p q )cosh(a x) + q - 8p q +
8p
+--R *
+--R +---------+
+--R | 2 2
+--R \|- q + p
+--R +
+--R 4 3 2 2 4 3 2
+--R (4p q - 4p q )sinh(a x) + (8p q - 8p q )cosh(a x)sinh(a
x)
+--R +
+--R 4 3 2 2 4 3 2 5
+--R (4p q - 4p q )cosh(a x) - 4p q + 12p q - 8p
+--R /
+--R 2 4 2 3
+--R q sinh(a x) + 4q cosh(a x)sinh(a x)
+--R +
+--R 2 2 2 2 2
+--R (6q cosh(a x) - 2q + 4p )sinh(a x)
+--R +
+--R 2 3 2 2 2
4
+--R (4q cosh(a x) + (- 4q + 8p )cosh(a x))sinh(a x) + q
cosh(a x)
+--R +
+--R 2 2 2 2
+--R (- 2q + 4p )cosh(a x) + q
+--R +
+--R +-------+
+--R +---------+ | 2 2
+--R | 2 2 - tanh(a x)\|q + p - p
+--R - \|- q + p log(-------------------------)
+--R +-------+
+--R | 2 2
+--R tanh(a x)\|q + p - p
+--R /
+--R +---------+ +-------+
+--R | 2 2 | 2 2
+--R 2a p\|- q + p \|q + p
+--R Type: Expression
Integer
+--E
+
+--S 79 14:556 Axiom cannot simplify this expression
+cc2:=aa.2-bb
+--R
+--R (4)
+--R +-------+
+--R +-------+ | 2 2
+--R | 2 2 - tanh(a x)\|q + p - p
+--R - \|q - p log(-------------------------)
+--R +-------+
+--R | 2 2
+--R tanh(a x)\|q + p - p
+--R +
+--R +-------+
+--R | 2 2
+--R 2\|q + p
+--R *
+--R atan
+--R 2 2 2 2 2 2
2
+--R (q sinh(a x) + 2q cosh(a x)sinh(a x) + q cosh(a x) - q +
2p )
+--R *
+--R +-------+
+--R | 2 2
+--R \|q - p
+--R /
+--R 2 3
+--R 2p q - 2p
+--R /
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R 2a p\|q - p \|q + p
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.557~~~~~$\displaystyle
@@ -510,7 +1461,7 @@ $$
<<*>>=
)clear all
---S 18 of 22
+--S 80 14:557 Axiom cannot compute this integral
aa:=integrate(x^m*sinh(a*x),x)
--R
--R
@@ -530,7 +1481,7 @@ $$
<<*>>=
)clear all
---S 19 of 22
+--S 81 14:558 Axiom cannot compute this integral
aa:=integrate(sinh(a*x)^n,x)
--R
--R
@@ -550,16 +1501,17 @@ $$
<<*>>=
)clear all
---S 20 of 22
-aa:=integrate(sinh(a*x)/a^n,x)
+--S 82 14:559 Axiom cannot compute this integral
+aa:=integrate(sinh(a*x)/x^n,x)
--R
---R 2 2
---R sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) + 1
---R (1) -------------------------------------------------
---R n
---R (2a sinh(a x) + 2a cosh(a x))a
+--R x
+--I ++ sinh(%T a)
+--I (3) | ---------- d%T
+--R ++ n
+--I %T
--R Type: Union(Expression
Integer,...)
--E
+
@
\section{\cite{1}:14.560~~~~~$\displaystyle
@@ -571,7 +1523,7 @@ $$
<<*>>=
)clear all
---S 21 of 22
+--S 83 14:560 Axiom cannot compute this integral
aa:=integrate(1/sinh(a*x)^n,x)
--R
--R
@@ -594,7 +1546,7 @@ $$
<<*>>=
)clear all
---S 22 of 22
+--S 84 14:561 Axiom cannot compute this integral
aa:=integrate(x/sinh(a*x)^n,x)
--R
--R
diff --git a/src/input/schaum28.input.pamphlet
b/src/input/schaum28.input.pamphlet
index b3fc3c6..5c68b06 100644
--- a/src/input/schaum28.input.pamphlet
+++ b/src/input/schaum28.input.pamphlet
@@ -18,7 +18,7 @@ $$
)set message auto off
)clear all
---S 1 of 28
+--S 1
aa:=integrate(cosh(a*x),x)
--R
--R
@@ -27,6 +27,22 @@ aa:=integrate(cosh(a*x),x)
--R a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 2
+bb:=sinh(a*x)/a
+--R
+--R sinh(a x)
+--R (2) ---------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 3 14:562 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.563~~~~~$\displaystyle
@@ -37,7 +53,7 @@ $$
<<*>>=
)clear all
---S 2 of 28
+--S 4
aa:=integrate(x*cosh(a*x),x)
--R
--R
@@ -47,6 +63,23 @@ aa:=integrate(x*cosh(a*x),x)
--R a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 5
+bb:=(x*sinh(a*x))/a-cosh(a*x)/a^2
+--R
+--R a x sinh(a x) - cosh(a x)
+--R (2) -------------------------
+--R 2
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 6 14:563 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.564~~~~~$\displaystyle
@@ -57,7 +90,7 @@ $$
<<*>>=
)clear all
---S 3 of 28
+--S 7
aa:=integrate(x^2*cosh(a*x),x)
--R
--R
@@ -68,6 +101,24 @@ aa:=integrate(x^2*cosh(a*x),x)
--R a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 8
+bb:=-(2*x*cosh(a*x))/a^2+(x^2/a+2/a^3)*sinh(a*x)
+--R
+--R 2 2
+--R (a x + 2)sinh(a x) - 2a x cosh(a x)
+--R (2) ------------------------------------
+--R 3
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 9 14:564 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.565~~~~~$\displaystyle
@@ -80,7 +131,7 @@ $$
<<*>>=
)clear all
---S 4 of 28
+--S 10 14:565 Axiom cannot compute this integral
aa:=integrate(cosh(a*x)/x,x)
--R
--R
@@ -100,7 +151,7 @@ $$
<<*>>=
)clear all
---S 5 of 28
+--S 11 14:566 Axiom cannot compute this integral
aa:=integrate(cosh(a*x)/x^2,x)
--R
--R
@@ -121,7 +172,7 @@ $$
<<*>>=
)clear all
---S 6 of 28
+--S 12
aa:=integrate(1/cosh(a*x),x)
--R
--R
@@ -130,6 +181,33 @@ aa:=integrate(1/cosh(a*x),x)
--R a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 13
+bb:=2/a*atan(%e^(a*x))
+--R
+--R a x
+--R 2atan(%e )
+--R (2) ------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 14
+cc:=aa-bb
+--R
+--R a x
+--R 2atan(sinh(a x) + cosh(a x)) - 2atan(%e )
+--R (3) -------------------------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 15 14:567 Schaums and Axiom agree
+dd:=complexNormalize cc
+--R
+--R (4) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.568~~~~~$\displaystyle
@@ -141,7 +219,7 @@ $$
<<*>>=
)clear all
---S 7 of 28
+--S 16 14:568 Axiom cannot compute this integral
aa:=integrate(x/cosh(a*x),x)
--R
--R
@@ -161,7 +239,7 @@ $$
<<*>>=
)clear all
---S 8 of 28
+--S 17
aa:=integrate(cosh(a*x)^2,x)
--R
--R
@@ -170,6 +248,42 @@ aa:=integrate(cosh(a*x)^2,x)
--R 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 18
+bb:=x/2+(sinh(a*x)*cosh(a*x))/2
+--R
+--R cosh(a x)sinh(a x) + x
+--R (2) ----------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 19
+cc:=aa-bb
+--R
+--R (- a + 1)cosh(a x)sinh(a x)
+--R (3) ---------------------------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 20
+sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y)))
+--R
+--I %B sinh(y + x) - %B sinh(y - x)
+--I (4) %B cosh(y)sinh(x) == -------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 21 14:569 SCHAUMS AND AXIOM DISAGREE?
+dd:=sinhcoshrule cc
+--R
+--R (- a + 1)sinh(2a x)
+--R (5) -------------------
+--R 4a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.570~~~~~$\displaystyle
@@ -180,7 +294,7 @@ $$
<<*>>=
)clear all
---S 9 of 28
+--S 22
aa:=integrate(x*cosh(a*x)^2,x)
--R
--R
@@ -191,6 +305,87 @@ aa:=integrate(x*cosh(a*x)^2,x)
--R 8a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 23
+bb:=x^2/4+(x*sinh(2*a*x))/(4*a)-cosh(2*a*x)/(8*a^2)
+--R
+--R 2 2
+--R 2a x sinh(2a x) - cosh(2a x) + 2a x
+--R (2) ------------------------------------
+--R 2
+--R 8a
+--R Type: Expression
Integer
+--E
+
+--S 24
+cc:=aa-bb
+--R
+--R (3)
+--R 2
+--R - 2a x sinh(2a x) - sinh(a x) + 4a x cosh(a x)sinh(a x) + cosh(2a x)
+--R +
+--R 2
+--R - cosh(a x)
+--R /
+--R 2
+--R 8a
+--R Type: Expression
Integer
+--E
+
+--S 25
+sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
+--R
+--R 2 cosh(2x) - 1
+--R (4) sinh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 26
+dd:=sinhsqrrule cc
+--R
+--R (5)
+--R 2
+--R - 4a x sinh(2a x) + 8a x cosh(a x)sinh(a x) + cosh(2a x) - 2cosh(a x) +
1
+--R
--------------------------------------------------------------------------
+--R 2
+--R 16a
+--R Type: Expression
Integer
+--E
+
+--S 27
+coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
+--R
+--R 2 cosh(2x) + 1
+--R (6) cosh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 28
+ee:=coshsqrrule dd
+--R
+--R - x sinh(2a x) + 2x cosh(a x)sinh(a x)
+--R (7) --------------------------------------
+--R 4a
+--R Type: Expression
Integer
+--E
+
+--S 29
+sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y)))
+--R
+--I %S sinh(y + x) - %S sinh(y - x)
+--I (8) %S cosh(y)sinh(x) == -------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 30 14:570 Schaums and Axiom agree
+ff:=sinhcoshrule ee
+--R
+--R (9) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.571~~~~~$\displaystyle
@@ -201,7 +396,7 @@ $$
<<*>>=
)clear all
---S 10 of 28
+--S 31
aa:=integrate(1/cosh(a*x)^2,x)
--R
--R
@@ -211,6 +406,176 @@ aa:=integrate(1/cosh(a*x)^2,x)
--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 32
+bb:=tanh(a*x)/a
+--R
+--R tanh(a x)
+--R (2) ---------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 33
+cc:=aa-bb
+--R
+--R 2 2
+--R (- sinh(a x) - 2cosh(a x)sinh(a x) - cosh(a x) - 1)tanh(a x) - 2
+--R (3) ------------------------------------------------------------------
+--R 2 2
+--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a
+--R Type: Expression
Integer
+--E
+
+--S 34
+tanhrule:=rule(tanh(x) == sinh(x)/cosh(x))
+--R
+--R sinh(x)
+--R (4) tanh(x) == -------
+--R cosh(x)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 35
+dd:=tanhrule cc
+--R
+--R (5)
+--R 3 2 2
+--R - sinh(a x) - 2cosh(a x)sinh(a x) + (- cosh(a x) - 1)sinh(a x)
+--R +
+--R - 2cosh(a x)
+--R /
+--R 2 2 3
+--R a cosh(a x)sinh(a x) + 2a cosh(a x) sinh(a x) + a cosh(a x) + a
cosh(a x)
+--R Type: Expression
Integer
+--E
+
+--S 36
+sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
+--R
+--R 2 cosh(2x) - 1
+--R (6) sinh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 37
+ee:=sinhsqrrule dd
+--R
+--R (7)
+--R 3 2
+--R - 2sinh(a x) + (- 2cosh(a x) - 2)sinh(a x) - 2cosh(a x)cosh(2a x)
+--R +
+--R - 2cosh(a x)
+--R /
+--R 2 3
+--R 4a cosh(a x) sinh(a x) + a cosh(a x)cosh(2a x) + 2a cosh(a x)
+--R +
+--R a cosh(a x)
+--R Type: Expression
Integer
+--E
+
+--S 38
+coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
+--R
+--R 2 cosh(2x) + 1
+--R (8) cosh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 39
+ff:=coshsqrrule ee
+--R
+--R (9)
+--R 3
+--R - 2sinh(a x) + (- cosh(2a x) - 3)sinh(a x) - 2cosh(a x)cosh(2a x)
+--R +
+--R - 2cosh(a x)
+--R /
+--R 3
+--R (2a cosh(2a x) + 2a)sinh(a x) + a cosh(a x)cosh(2a x) + 2a cosh(a x)
+--R +
+--R a cosh(a x)
+--R Type: Expression
Integer
+--E
+
+--S 40
+sinhcuberule:=rule(sinh(x)^3 == 1/4*sinh(3*x)-3/4*sinh(x))
+--R
+--R 3 sinh(3x) - 3sinh(x)
+--R (10) sinh(x) == -------------------
+--R 4
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 41
+gg:=sinhcuberule ff
+--R
+--R (11)
+--R - sinh(3a x) + (- 2cosh(2a x) - 3)sinh(a x) - 4cosh(a x)cosh(2a x)
+--R +
+--R - 4cosh(a x)
+--R /
+--R
3
+--R (4a cosh(2a x) + 4a)sinh(a x) + 2a cosh(a x)cosh(2a x) + 4a cosh(a x)
+--R +
+--R 2a cosh(a x)
+--R Type: Expression
Integer
+--E
+
+--S 42
+coshcuberule:=rule(cosh(x)^3 == 1/4*cosh(3*x)-3/4*cosh(x))
+--R
+--R 3 cosh(3x) - 3cosh(x)
+--R (12) cosh(x) == -------------------
+--R 4
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 43
+hh:=coshcuberule gg
+--R
+--R (13)
+--R - sinh(3a x) + (- 2cosh(2a x) - 3)sinh(a x) - 4cosh(a x)cosh(2a x)
+--R +
+--R - 4cosh(a x)
+--R /
+--R (4a cosh(2a x) + 4a)sinh(a x) + a cosh(3a x) + 2a cosh(a x)cosh(2a x)
+--R +
+--R - a cosh(a x)
+--R Type: Expression
Integer
+--E
+
+--S 44
+coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y)))
+--R
+--I %U cosh(y + x) + %U cosh(y - x)
+--I (14) %U cosh(x)cosh(y) == -------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 45
+ii:=coshcoshrule hh
+--R
+--R (15)
+--R - sinh(3a x) + (- 2cosh(2a x) - 3)sinh(a x) - 2cosh(3a x) - 6cosh(a x)
+--R ----------------------------------------------------------------------
+--R (4a cosh(2a x) + 4a)sinh(a x) + 2a cosh(3a x)
+--R Type: Expression
Integer
+--E
+
+--S 46 14:571 SCHAUMS AND AXIOM DISAGREE?
+jj:=complexNormalize ii
+--R
+--R a x 2
+--R - 2(%e ) - 2
+--R (16) ---------------
+--R a x 2
+--R 2a (%e ) - a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.572~~~~~$\displaystyle
@@ -221,7 +586,7 @@ $$
<<*>>=
)clear all
---S 11 of 28
+--S 47
aa:=integrate(cosh(a*x)*cosh(p*x),x)
--R
--R
@@ -231,6 +596,106 @@ aa:=integrate(cosh(a*x)*cosh(p*x),x)
--R (p - a )sinh(a x) + (- p + a )cosh(a x)
--R Type: Union(Expression
Integer,...)
--E
+
+--S 48
+bb:=(sinh(a-p)*x)/(2*(a-p))+(sinh(a+p)*x)/(2*(a+p))
+--R
+--R (p - a)x sinh(p + a) + (p + a)x sinh(p - a)
+--R (2) -------------------------------------------
+--R 2 2
+--R 2p - 2a
+--R Type: Expression
Integer
+--E
+
+--S 49
+cc:=aa-bb
+--R
+--R (3)
+--R - 2p cosh(a x)sinh(p x)
+--R +
+--R 2
+--R ((- p + a)x sinh(p + a) + (- p - a)x sinh(p - a))sinh(a x)
+--R +
+--R 2
+--R 2a cosh(p x)sinh(a x) + (p - a)x cosh(a x) sinh(p + a)
+--R +
+--R 2
+--R (p + a)x cosh(a x) sinh(p - a)
+--R /
+--R 2 2 2 2 2 2
+--R (2p - 2a )sinh(a x) + (- 2p + 2a )cosh(a x)
+--R Type: Expression
Integer
+--E
+
+--S 50
+sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
+--R
+--R 2 cosh(2x) - 1
+--R (4) sinh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 51
+dd:=sinhsqrrule cc
+--R
+--R (5)
+--R - 4p cosh(a x)sinh(p x) + 4a cosh(p x)sinh(a x)
+--R +
+--R 2
+--R ((- p + a)x cosh(2a x) + (2p - 2a)x cosh(a x) + (p - a)x)sinh(p + a)
+--R +
+--R 2
+--R ((- p - a)x cosh(2a x) + (2p + 2a)x cosh(a x) + (p + a)x)sinh(p - a)
+--R /
+--R 2 2 2 2 2 2 2
+--R (2p - 2a )cosh(2a x) + (- 4p + 4a )cosh(a x) - 2p + 2a
+--R Type: Expression
Integer
+--E
+
+--S 52
+coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
+--R
+--R 2 cosh(2x) + 1
+--R (6) cosh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 53
+ee:=coshsqrrule dd
+--R
+--R (7)
+--R 2p cosh(a x)sinh(p x) - 2a cosh(p x)sinh(a x) + (- p + a)x sinh(p +
a)
+--R +
+--R (- p - a)x sinh(p - a)
+--R /
+--R 2 2
+--R 2p - 2a
+--R Type: Expression
Integer
+--E
+
+--S 54
+sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y)))
+--R
+--I %V sinh(y + x) - %V sinh(y - x)
+--I (8) %V cosh(y)sinh(x) == -------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 55 14:572 Schaums and Axiom agree
+ff:=sinhcoshrule ee
+--R
+--R (9)
+--R (p - a)sinh((p + a)x) + (p + a)sinh((p - a)x) + (- p + a)x sinh(p +
a)
+--R +
+--R (- p - a)x sinh(p - a)
+--R /
+--R 2 2
+--R 2p - 2a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.573~~~~~$\displaystyle
@@ -241,7 +706,7 @@ $$
<<*>>=
)clear all
---S 12 of 28
+--S 56
aa:=integrate(cosh(a*x)*sin(p*x),x)
--R
--R
@@ -258,6 +723,70 @@ aa:=integrate(cosh(a*x)*sin(p*x),x)
--R (2p + 2a )sinh(a x) + (2p + 2a )cosh(a x)
--R Type: Union(Expression
Integer,...)
--E
+
+--S 57
+bb:=(a*sinh(a*x)*sin(p*x)-p*cosh(a*x)*cos(p*x))/(a^2+p^2)
+--R
+--R a sin(p x)sinh(a x) - p cos(p x)cosh(a x)
+--R (2) -----------------------------------------
+--R 2 2
+--R p + a
+--R Type: Expression
Integer
+--E
+
+--S 58
+cc:=aa-bb
+--R
+--R (3)
+--R 2 2
+--R (- a sin(p x) - p cos(p x))sinh(a x) + (a cosh(a x) - a)sin(p x)
+--R +
+--R 2
+--R p cos(p x)cosh(a x) - p cos(p x)
+--R /
+--R 2 2 2 2
+--R (2p + 2a )sinh(a x) + (2p + 2a )cosh(a x)
+--R Type: Expression
Integer
+--E
+
+--S 59
+coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
+--R
+--R 2 cosh(2x) + 1
+--R (4) cosh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 60
+dd:=coshsqrrule cc
+--R
+--R (5)
+--R 2
+--R (- 2a sin(p x) - 2p cos(p x))sinh(a x) + (a cosh(2a x) - a)sin(p x)
+--R +
+--R p cos(p x)cosh(2a x) - p cos(p x)
+--R /
+--R 2 2 2 2
+--R (4p + 4a )sinh(a x) + (4p + 4a )cosh(a x)
+--R Type: Expression
Integer
+--E
+
+--S 61
+sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
+--R
+--R 2 cosh(2x) - 1
+--R (6) sinh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 62 14:573 Schaums and Axiom agree
+ee:=sinhsqrrule dd
+--R
+--R (7) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.574~~~~~$\displaystyle
@@ -268,7 +797,7 @@ $$
<<*>>=
)clear all
---S 13 of 28
+--S 63
aa:=integrate(cosh(a*x)*cos(p*x),x)
--R
--R
@@ -285,6 +814,70 @@ aa:=integrate(cosh(a*x)*cos(p*x),x)
--R (2p + 2a )sinh(a x) + (2p + 2a )cosh(a x)
--R Type: Union(Expression
Integer,...)
--E
+
+--S 64
+bb:=(a*sinh(a*x)*cos(p*x)+p*cosh(a*x)*sin(p*x))/(a^2+p^2)
+--R
+--R a cos(p x)sinh(a x) + p cosh(a x)sin(p x)
+--R (2) -----------------------------------------
+--R 2 2
+--R p + a
+--R Type: Expression
Integer
+--E
+
+--S 65
+cc:=aa-bb
+--R
+--R (3)
+--R 2 2
+--R (p sin(p x) - a cos(p x))sinh(a x) + (- p cosh(a x) + p)sin(p x)
+--R +
+--R 2
+--R a cos(p x)cosh(a x) - a cos(p x)
+--R /
+--R 2 2 2 2
+--R (2p + 2a )sinh(a x) + (2p + 2a )cosh(a x)
+--R Type: Expression
Integer
+--E
+
+--S 66
+coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
+--R
+--R 2 cosh(2x) + 1
+--R (4) cosh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 67
+dd:=coshsqrrule cc
+--R
+--R (5)
+--R 2
+--R (2p sin(p x) - 2a cos(p x))sinh(a x) + (- p cosh(2a x) + p)sin(p x)
+--R +
+--R a cos(p x)cosh(2a x) - a cos(p x)
+--R /
+--R 2 2 2 2
+--R (4p + 4a )sinh(a x) + (4p + 4a )cosh(a x)
+--R Type: Expression
Integer
+--E
+
+--S 68
+sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
+--R
+--R 2 cosh(2x) - 1
+--R (6) sinh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 69 14:574 Schaums and Axiom agree
+ee:=sinhsqrrule dd
+--R
+--R (7) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.575~~~~~$\displaystyle
@@ -295,7 +888,7 @@ $$
<<*>>=
)clear all
---S 14 of 28
+--S 70
aa:=integrate(1/(cosh(a*x)+1),x)
--R
--R
@@ -304,6 +897,112 @@ aa:=integrate(1/(cosh(a*x)+1),x)
--R a sinh(a x) + a cosh(a x) + a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 71
+bb:=1/a*tanh((a*x)/2)
+--R
+--R a x
+--R tanh(---)
+--R 2
+--R (2) ---------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 72
+cc:=aa-bb
+--R
+--R a x
+--R (- sinh(a x) - cosh(a x) - 1)tanh(---) - 2
+--R 2
+--R (3) ------------------------------------------
+--R a sinh(a x) + a cosh(a x) + a
+--R Type: Expression
Integer
+--E
+
+--S 73
+tanhrule:=rule(tanh(x) == sinh(x)/cosh(x))
+--R
+--R sinh(x)
+--R (4) tanh(x) == -------
+--R cosh(x)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 74
+dd:=tanhrule cc
+--R
+--R a x a x a x
+--R - sinh(---)sinh(a x) + (- cosh(a x) - 1)sinh(---) - 2cosh(---)
+--R 2 2 2
+--R (5) --------------------------------------------------------------
+--R a x a x a x
+--R a cosh(---)sinh(a x) + a cosh(---)cosh(a x) + a cosh(---)
+--R 2 2 2
+--R Type: Expression
Integer
+--E
+
+--S 75
+sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y)))
+--R
+--I %B sinh(y + x) - %B sinh(y - x)
+--I (6) %B cosh(y)sinh(x) == -------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 76
+ee:=sinhcoshrule dd
+--R
+--R 3a x a x a x a x
+--R - sinh(----) - 2sinh(---)sinh(a x) - sinh(---) - 4cosh(---)
+--R 2 2 2 2
+--R (7) -----------------------------------------------------------------
+--R 3a x a x a x a x
+--R a sinh(----) + a sinh(---) + 2a cosh(---)cosh(a x) + 2a cosh(---)
+--R 2 2 2 2
+--R Type: Expression
Integer
+--E
+
+--S 77
+sinhsinhrule:=rule(sinh(x)*sinh(y)==1/2*(cosh(x+y)-cosh(x-y)))
+--R
+--I %B cosh(y + x) - %B cosh(y - x)
+--I (8) %B sinh(x)sinh(y) == -------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 78
+ff:=sinhsinhrule ee
+--R
+--R 3a x a x 3a x a x
+--R - sinh(----) - sinh(---) - cosh(----) - 3cosh(---)
+--R 2 2 2 2
+--R (9) -----------------------------------------------------------------
+--R 3a x a x a x a x
+--R a sinh(----) + a sinh(---) + 2a cosh(---)cosh(a x) + 2a cosh(---)
+--R 2 2 2 2
+--R Type: Expression
Integer
+--E
+
+--S 79
+coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y)))
+--R
+--I %BC cosh(y + x) + %BC cosh(y - x)
+--I (10) %BC cosh(x)cosh(y) == ---------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 80 14:575 Schaums and Axiom differ by a constant
+gg:=coshcoshrule ff
+--R
+--R 1
+--R (11) - -
+--R a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.576~~~~~$\displaystyle
@@ -314,7 +1013,7 @@ $$
<<*>>=
)clear all
---S 15 of 28
+--S 81
aa:=integrate(1/(cosh(a*x)-1),x)
--R
--R
@@ -323,6 +1022,112 @@ aa:=integrate(1/(cosh(a*x)-1),x)
--R a sinh(a x) + a cosh(a x) - a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 82
+bb:=-1/a*coth((a*x)/2)
+--R
+--R a x
+--R coth(---)
+--R 2
+--R (2) - ---------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 83
+cc:=aa-bb
+--R
+--R a x a x
+--R coth(---)sinh(a x) + (cosh(a x) - 1)coth(---) - 2
+--R 2 2
+--R (3) -------------------------------------------------
+--R a sinh(a x) + a cosh(a x) - a
+--R Type: Expression
Integer
+--E
+
+--S 84
+cothrule:=rule(coth(x) == cosh(x)/sinh(x))
+--R
+--R cosh(x)
+--R (4) coth(x) == -------
+--R sinh(x)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 85
+dd:=cothrule cc
+--R
+--R a x a x a x a x
+--R cosh(---)sinh(a x) - 2sinh(---) + cosh(---)cosh(a x) - cosh(---)
+--R 2 2 2 2
+--R (5) ----------------------------------------------------------------
+--R a x a x
+--R a sinh(---)sinh(a x) + (a cosh(a x) - a)sinh(---)
+--R 2 2
+--R Type: Expression
Integer
+--E
+
+--S 86
+sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y)))
+--R
+--I %BD sinh(y + x) - %BD sinh(y - x)
+--I (6) %BD cosh(y)sinh(x) == ---------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 87
+ee:=sinhcoshrule dd
+--R
+--R 3a x a x a x a x
+--R sinh(----) - 3sinh(---) + 2cosh(---)cosh(a x) - 2cosh(---)
+--R 2 2 2 2
+--R (7) ----------------------------------------------------------
+--R 3a x a x a x
+--R a sinh(----) + 2a sinh(---)sinh(a x) - 3a sinh(---)
+--R 2 2 2
+--R Type: Expression
Integer
+--E
+
+--S 88
+sinhsinhrule:=rule(sinh(x)*sinh(y)==1/2*(cosh(x+y)-cosh(x-y)))
+--R
+--I %BE cosh(y + x) - %BE cosh(y - x)
+--I (8) %BE sinh(x)sinh(y) == ---------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 89
+ff:=sinhsinhrule ee
+--R
+--R 3a x a x a x a x
+--R sinh(----) - 3sinh(---) + 2cosh(---)cosh(a x) - 2cosh(---)
+--R 2 2 2 2
+--R (9) ----------------------------------------------------------
+--R 3a x a x 3a x a x
+--R a sinh(----) - 3a sinh(---) + a cosh(----) - a cosh(---)
+--R 2 2 2 2
+--R Type: Expression
Integer
+--E
+
+--S 90
+coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y)))
+--R
+--I %BF cosh(y + x) + %BF cosh(y - x)
+--I (10) %BF cosh(x)cosh(y) == ---------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 91 14:576 Schaums and Axiom differ by a constant
+gg:=coshcoshrule ff
+--R
+--R 1
+--R (11) -
+--R a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.577~~~~~$\displaystyle
@@ -333,7 +1138,7 @@ $$
<<*>>=
)clear all
---S 16 of 28
+--S 92
aa:=integrate(x/(cosh(a*x)+1),x)
--R
--R
@@ -346,6 +1151,181 @@ aa:=integrate(x/(cosh(a*x)+1),x)
--R a sinh(a x) + a cosh(a x) + a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 93
+bb:=x/a*tanh((a*x)/2)-2/a^2*log(cosh((a*x)/2))
+--R
+--R a x a x
+--R - 2log(cosh(---)) + a x tanh(---)
+--R 2 2
+--R (2) ---------------------------------
+--R 2
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 94
+cc:=aa-bb
+--R
+--R (3)
+--R (- 2sinh(a x) - 2cosh(a x) - 2)log(sinh(a x) + cosh(a x) + 1)
+--R +
+--R a x
+--R (2sinh(a x) + 2cosh(a x) + 2)log(cosh(---))
+--R 2
+--R +
+--R a x
+--R (- a x sinh(a x) - a x cosh(a x) - a x)tanh(---) + 2a x sinh(a x)
+--R 2
+--R +
+--R 2a x cosh(a x)
+--R /
+--R 2 2 2
+--R a sinh(a x) + a cosh(a x) + a
+--R Type: Expression
Integer
+--E
+
+--S 95
+tanhrule:=rule(tanh(x) == sinh(x)/cosh(x))
+--R
+--R sinh(x)
+--R (4) tanh(x) == -------
+--R cosh(x)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 96
+dd:=tanhrule cc
+--R
+--R (5)
+--R a x a x a x
+--R (- 2cosh(---)sinh(a x) - 2cosh(---)cosh(a x) - 2cosh(---))
+--R 2 2 2
+--R *
+--R log(sinh(a x) + cosh(a x) + 1)
+--R +
+--R a x a x a x a x
+--R (2cosh(---)sinh(a x) + 2cosh(---)cosh(a x) +
2cosh(---))log(cosh(---))
+--R 2 2 2 2
+--R +
+--R a x a x
+--R (- a x sinh(---) + 2a x cosh(---))sinh(a x)
+--R 2 2
+--R +
+--R a x a x
+--R (- a x cosh(a x) - a x)sinh(---) + 2a x cosh(---)cosh(a x)
+--R 2 2
+--R /
+--R 2 a x 2 a x 2 a x
+--R a cosh(---)sinh(a x) + a cosh(---)cosh(a x) + a cosh(---)
+--R 2 2 2
+--R Type: Expression
Integer
+--E
+
+--S 97
+coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y)))
+--R
+--I %BG cosh(y + x) + %BG cosh(y - x)
+--I (6) %BG cosh(x)cosh(y) == ---------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 98
+ee:=coshcoshrule dd
+--R
+--R (7)
+--R a x 3a x a x
+--R (- 4cosh(---)sinh(a x) - 2cosh(----) - 6cosh(---))
+--R 2 2 2
+--R *
+--R log(sinh(a x) + cosh(a x) + 1)
+--R +
+--R a x 3a x a x a x
+--R (4cosh(---)sinh(a x) + 2cosh(----) + 6cosh(---))log(cosh(---))
+--R 2 2 2 2
+--R +
+--R a x a x
+--R (- 2a x sinh(---) + 4a x cosh(---))sinh(a x)
+--R 2 2
+--R +
+--R a x 3a x a x
+--R (- 2a x cosh(a x) - 2a x)sinh(---) + 2a x cosh(----) + 2a x cosh(---)
+--R 2 2 2
+--R /
+--R 2 a x 2 3a x 2 a x
+--R 2a cosh(---)sinh(a x) + a cosh(----) + 3a cosh(---)
+--R 2 2 2
+--R Type: Expression
Integer
+--E
+
+--S 99
+sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y)))
+--R
+--I %BH sinh(y + x) - %BH sinh(y - x)
+--I (8) %BH cosh(y)sinh(x) == ---------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 100
+ff:=sinhcoshrule ee
+--R
+--R (9)
+--R 3a x a x 3a x a x
+--R (- 2sinh(----) - 2sinh(---) - 2cosh(----) - 6cosh(---))
+--R 2 2 2 2
+--R *
+--R log(sinh(a x) + cosh(a x) + 1)
+--R +
+--R 3a x a x 3a x a x a x
+--R (2sinh(----) + 2sinh(---) + 2cosh(----) + 6cosh(---))log(cosh(---))
+--R 2 2 2 2 2
+--R +
+--R 3a x a x a x
+--R a x sinh(----) - 2a x sinh(---)sinh(a x) + a x sinh(---)
+--R 2 2 2
+--R +
+--R 3a x a x
+--R 2a x cosh(----) + 2a x cosh(---)
+--R 2 2
+--R /
+--R 2 3a x 2 a x 2 3a x 2 a x
+--R a sinh(----) + a sinh(---) + a cosh(----) + 3a cosh(---)
+--R 2 2 2 2
+--R Type: Expression
Integer
+--E
+
+--S 101
+sinhsinhrule:=rule(sinh(x)*sinh(y)==1/2*(cosh(x+y)-cosh(x-y)))
+--R
+--I %BI cosh(y + x) - %BI cosh(y - x)
+--I (10) %BI sinh(x)sinh(y) == ---------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 102
+gg:=sinhsinhrule ff
+--R
+--R a x
+--R - 2log(sinh(a x) + cosh(a x) + 1) + 2log(cosh(---)) + a x
+--R 2
+--R (11) ---------------------------------------------------------
+--R 2
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 103 14:577 Schaums and Axiom differ by a constant
+complexNormalize gg
+--R
+--R 2log(2)
+--R (12) - -------
+--R 2
+--R a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.578~~~~~$\displaystyle
@@ -356,7 +1336,7 @@ $$
<<*>>=
)clear all
---S 17 of 28
+--S 104
aa:=integrate(x/(cosh(a*x)-1),x)
--R
--R
@@ -369,6 +1349,181 @@ aa:=integrate(x/(cosh(a*x)-1),x)
--R a sinh(a x) + a cosh(a x) - a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 105
+bb:=-x/a*coth((a*x)/2)+2/a^2*log(sinh((a*x)/2))
+--R
+--R a x a x
+--R 2log(sinh(---)) - a x coth(---)
+--R 2 2
+--R (2) -------------------------------
+--R 2
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 106
+cc:=aa-bb
+--R
+--R (3)
+--R (2sinh(a x) + 2cosh(a x) - 2)log(sinh(a x) + cosh(a x) - 1)
+--R +
+--R a x
+--R (- 2sinh(a x) - 2cosh(a x) + 2)log(sinh(---))
+--R 2
+--R +
+--R a x a x
+--R (a x coth(---) - 2a x)sinh(a x) + (a x cosh(a x) - a x)coth(---)
+--R 2 2
+--R +
+--R - 2a x cosh(a x)
+--R /
+--R 2 2 2
+--R a sinh(a x) + a cosh(a x) - a
+--R Type: Expression
Integer
+--E
+
+--S 107
+cothrule:=rule(coth(x) == cosh(x)/sinh(x))
+--R
+--R cosh(x)
+--R (4) coth(x) == -------
+--R sinh(x)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 108
+dd:=cothrule cc
+--R
+--R (5)
+--R a x a x
+--R (2sinh(---)sinh(a x) + (2cosh(a x) - 2)sinh(---))
+--R 2 2
+--R *
+--R log(sinh(a x) + cosh(a x) - 1)
+--R +
+--R a x a x a x
+--R (- 2sinh(---)sinh(a x) + (- 2cosh(a x) + 2)sinh(---))log(sinh(---))
+--R 2 2 2
+--R +
+--R a x a x a x
+--R (- 2a x sinh(---) + a x cosh(---))sinh(a x) - 2a x cosh(a x)sinh(---)
+--R 2 2 2
+--R +
+--R a x a x
+--R a x cosh(---)cosh(a x) - a x cosh(---)
+--R 2 2
+--R /
+--R 2 a x 2 2 a x
+--R a sinh(---)sinh(a x) + (a cosh(a x) - a )sinh(---)
+--R 2 2
+--R Type: Expression
Integer
+--E
+
+--S 109
+sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y)))
+--R
+--I %BJ sinh(y + x) - %BJ sinh(y - x)
+--I (6) %BJ cosh(y)sinh(x) == ---------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 110
+ee:=sinhcoshrule dd
+--R
+--R (7)
+--R 3a x a x a x
+--R (2sinh(----) + 4sinh(---)sinh(a x) - 6sinh(---))
+--R 2 2 2
+--R *
+--R log(sinh(a x) + cosh(a x) - 1)
+--R +
+--R 3a x a x a x a x
+--R (- 2sinh(----) - 4sinh(---)sinh(a x) + 6sinh(---))log(sinh(---))
+--R 2 2 2 2
+--R +
+--R 3a x a x a x
+--R - a x sinh(----) - 4a x sinh(---)sinh(a x) + 3a x sinh(---)
+--R 2 2 2
+--R +
+--R a x a x
+--R 2a x cosh(---)cosh(a x) - 2a x cosh(---)
+--R 2 2
+--R /
+--R 2 3a x 2 a x 2 a x
+--R a sinh(----) + 2a sinh(---)sinh(a x) - 3a sinh(---)
+--R 2 2 2
+--R Type: Expression
Integer
+--E
+
+--S 111
+sinhsinhrule:=rule(sinh(x)*sinh(y)==1/2*(cosh(x+y)-cosh(x-y)))
+--R
+--I %BK cosh(y + x) - %BK cosh(y - x)
+--I (8) %BK sinh(x)sinh(y) == ---------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 112
+ff:=sinhsinhrule ee
+--R
+--R (9)
+--R 3a x a x 3a x a x
+--R (2sinh(----) - 6sinh(---) + 2cosh(----) - 2cosh(---))
+--R 2 2 2 2
+--R *
+--R log(sinh(a x) + cosh(a x) - 1)
+--R +
+--R 3a x a x 3a x a x a x
+--R (- 2sinh(----) + 6sinh(---) - 2cosh(----) + 2cosh(---))log(sinh(---))
+--R 2 2 2 2 2
+--R +
+--R 3a x a x 3a x
+--R - a x sinh(----) + 3a x sinh(---) - 2a x cosh(----)
+--R 2 2 2
+--R +
+--R a x
+--R 2a x cosh(---)cosh(a x)
+--R 2
+--R /
+--R 2 3a x 2 a x 2 3a x 2 a x
+--R a sinh(----) - 3a sinh(---) + a cosh(----) - a cosh(---)
+--R 2 2 2 2
+--R Type: Expression
Integer
+--E
+
+--S 113
+coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y)))
+--R
+--I %BL cosh(y + x) + %BL cosh(y - x)
+--I (10) %BL cosh(x)cosh(y) == ---------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 114
+gg:=coshcoshrule ff
+--R
+--R a x
+--R 2log(sinh(a x) + cosh(a x) - 1) - 2log(sinh(---)) - a x
+--R 2
+--R (11) -------------------------------------------------------
+--R 2
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 115 14:578 Schaums and Axiom differ by a constant
+hh:=complexNormalize gg
+--R
+--R 2log(2)
+--R (12) -------
+--R 2
+--R a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.579~~~~~$\displaystyle
@@ -379,7 +1534,7 @@ $$
<<*>>=
)clear all
---S 18 of 28
+--S 116
aa:=integrate(1/(cosh(a*x)+1)^2,x)
--R
--R
@@ -396,6 +1551,59 @@ aa:=integrate(1/(cosh(a*x)+1)^2,x)
--R 9a cosh(a x) + 9a cosh(a x) + 3a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 117
+bb:=1/(2*a)*tanh((a*x)/2)-1/(6*a)*tanh((a*x)/2)^3
+--R
+--R a x 3 a x
+--R - tanh(---) + 3tanh(---)
+--R 2 2
+--R (2) -------------------------
+--R 6a
+--R Type: Expression
Integer
+--E
+
+--S 118 14:579 Axiom cannot compute this integral
+cc:=aa-bb
+--R
+--R (3)
+--R 3 2
+--R sinh(a x) + (3cosh(a x) + 3)sinh(a x)
+--R +
+--R 2 3
2
+--R (3cosh(a x) + 6cosh(a x) + 3)sinh(a x) + cosh(a x) + 3cosh(a x)
+--R +
+--R 3cosh(a x) + 1
+--R *
+--R a x 3
+--R tanh(---)
+--R 2
+--R +
+--R 3 2
+--R - 3sinh(a x) + (- 9cosh(a x) - 9)sinh(a x)
+--R +
+--R 2 3
+--R (- 9cosh(a x) - 18cosh(a x) - 9)sinh(a x) - 3cosh(a x)
+--R +
+--R 2
+--R - 9cosh(a x) - 9cosh(a x) - 3
+--R *
+--R a x
+--R tanh(---)
+--R 2
+--R +
+--R - 12sinh(a x) - 12cosh(a x) - 4
+--R /
+--R 3 2
+--R 6a sinh(a x) + (18a cosh(a x) + 18a)sinh(a x)
+--R +
+--R 2 3
+--R (18a cosh(a x) + 36a cosh(a x) + 18a)sinh(a x) + 6a cosh(a x)
+--R +
+--R 2
+--R 18a cosh(a x) + 18a cosh(a x) + 6a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.580~~~~~$\displaystyle
@@ -406,7 +1614,7 @@ $$
<<*>>=
)clear all
---S 19 of 28
+--S 119
aa:=integrate(1/(cosh(a*x)-1)^2,x)
--R
--R
@@ -423,6 +1631,58 @@ aa:=integrate(1/(cosh(a*x)-1)^2,x)
--R - 9a cosh(a x) + 9a cosh(a x) - 3a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 120
+bb:=1/(2*a)*coth((a*x)/2)-1/(6*a)*coth((a*x)/2)^3
+--R
+--R a x 3 a x
+--R - coth(---) + 3coth(---)
+--R 2 2
+--R (2) -------------------------
+--R 6a
+--R Type: Expression
Integer
+--E
+
+--S 121 14:580 Axiom cannot compute this integral
+cc:=aa-bb
+--R
+--R (3)
+--R a x 3 a x 3
+--R (coth(---) - 3coth(---))sinh(a x)
+--R 2 2
+--R +
+--R a x 3 a x 2
+--R ((3cosh(a x) - 3)coth(---) + (- 9cosh(a x) + 9)coth(---))sinh(a x)
+--R 2 2
+--R +
+--R 2 a x 3
+--R (3cosh(a x) - 6cosh(a x) + 3)coth(---)
+--R 2
+--R +
+--R 2 a x
+--R (- 9cosh(a x) + 18cosh(a x) - 9)coth(---) - 12
+--R 2
+--R *
+--R sinh(a x)
+--R +
+--R 3 2 a x 3
+--R (cosh(a x) - 3cosh(a x) + 3cosh(a x) - 1)coth(---)
+--R 2
+--R +
+--R 3 2 a x
+--R (- 3cosh(a x) + 9cosh(a x) - 9cosh(a x) + 3)coth(---) - 12cosh(a x)
+ 4
+--R 2
+--R /
+--R 3 2
+--R 6a sinh(a x) + (18a cosh(a x) - 18a)sinh(a x)
+--R +
+--R 2 3
+--R (18a cosh(a x) - 36a cosh(a x) + 18a)sinh(a x) + 6a cosh(a x)
+--R +
+--R 2
+--R - 18a cosh(a x) + 18a cosh(a x) - 6a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.581~~~~~$\displaystyle
@@ -442,7 +1702,7 @@ $$
<<*>>=
)clear all
---S 20 of 28
+--S 122
aa:=integrate(1/(p+q*cosh(a*x)),x)
--R
--R
@@ -483,6 +1743,159 @@ aa:=integrate(1/(p+q*cosh(a*x)),x)
--R a\|q - p
--R Type: Union(List Expression
Integer,...)
--E
+
+--S 123
+bb1:=2/(a*sqrt(q^2-p^2))*atan((q*%e^(a*x)+p)/sqrt(q^2-p^2))
+--R
+--R a x
+--R q %e + p
+--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 124
+bb2:=1/(a*sqrt(p^2-q^2))*log((q*%e^(a*x)+p-sqrt(p^2-q^2))/(q*%e^(a*x)+p+sqrt(p^2-q^2)))
+--R
+--R +---------+
+--R | 2 2 a x
+--R - \|- q + p + q %e + p
+--R log(----------------------------)
+--R +---------+
+--R | 2 2 a x
+--R \|- q + p + q %e + p
+--R (3) ---------------------------------
+--R +---------+
+--R | 2 2
+--R a\|- q + p
+--R Type: Expression
Integer
+--E
+
+--S 125
+cc1:=aa.1-bb1
+--R
+--R (4)
+--R +-------+
+--R | 2 2
+--R \|q - p
+--R *
+--R log
+--R 2 2 2 2
2
+--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q
cosh(a x)
+--R +
+--R 2 2
+--R 2p q cosh(a x) - q + 2p
+--R *
+--R +---------+
+--R | 2 2
+--R \|- q + p
+--R +
+--R 3 2 3 2 2 3
+--R (2q - 2p q)sinh(a x) + (2q - 2p q)cosh(a x) + 2p q - 2p
+--R /
+--R 2 2
+--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x)
+--R +
+--R 2p cosh(a x) + q
+--R +
+--R +---------+ a x
+--R | 2 2 q %e + p
+--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 126
+cc2:=aa.2-bb1
+--R
+--R +-------+
+--R | 2 2 a x
+--R (q sinh(a x) + q cosh(a x) + p)\|q - p q %e + p
+--R 2atan(-----------------------------------------) -
2atan(-----------)
+--R 2 2 +-------+
+--R q - p | 2 2
+--R \|q - p
+--R (5)
---------------------------------------------------------------------
+--R +-------+
+--R | 2 2
+--R a\|q - p
+--R Type: Expression
Integer
+--E
+
+--S 127
+cc3:=aa.1-bb2
+--R
+--R (6)
+--R log
+--R 2 2 2 2
2
+--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a
x)
+--R +
+--R 2 2
+--R 2p q cosh(a x) - q + 2p
+--R *
+--R +---------+
+--R | 2 2
+--R \|- q + p
+--R +
+--R 3 2 3 2 2 3
+--R (2q - 2p q)sinh(a x) + (2q - 2p q)cosh(a x) + 2p q - 2p
+--R /
+--R 2 2
+--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x)
+--R +
+--R 2p cosh(a x) + q
+--R +
+--R +---------+
+--R | 2 2 a x
+--R - \|- q + p + q %e + p
+--R - log(----------------------------)
+--R +---------+
+--R | 2 2 a x
+--R \|- q + p + q %e + p
+--R /
+--R +---------+
+--R | 2 2
+--R a\|- q + p
+--R Type: Expression
Integer
+--E
+
+--S 128 14:581 Axiom cannot simplify this expression
+cc4:=aa.2-bb2
+--R
+--R (7)
+--R +---------+
+--R +-------+ | 2 2 a x
+--R | 2 2 - \|- q + p + q %e + p
+--R - \|q - p log(----------------------------)
+--R +---------+
+--R | 2 2 a x
+--R \|- q + p + q %e + p
+--R +
+--R +-------+
+--R +---------+ | 2 2
+--R | 2 2 (q sinh(a x) + q cosh(a x) + p)\|q - p
+--R 2\|- q + p atan(-----------------------------------------)
+--R 2 2
+--R q - p
+--R /
+--R +---------+ +-------+
+--R | 2 2 | 2 2
+--R a\|- q + p \|q - p
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.582~~~~~$\displaystyle
@@ -494,7 +1907,7 @@ $$
<<*>>=
)clear all
---S 21 of 28
+--S 129
aa:=integrate(1/(p+q*cosh(a*x))^2,x)
--R
--R
@@ -575,6 +1988,408 @@ aa:=integrate(1/(p+q*cosh(a*x))^2,x)
--R ]
--R Type: Union(List Expression
Integer,...)
--E
+
+--S 130
+t1:=integrate(1/(p+q*cosh(a*x)),x)
+--R
+--R (2)
+--R [
+--R log
+--R 2 2 2 2
2
+--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a
x)
+--R +
+--R 2 2
+--R 2p q cosh(a x) - q + 2p
+--R *
+--R +---------+
+--R | 2 2
+--R \|- q + p
+--R +
+--R 3 2 3 2 2 3
+--R (2q - 2p q)sinh(a x) + (2q - 2p q)cosh(a x) + 2p q - 2p
+--R /
+--R 2 2
+--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x)
+--R +
+--R 2p cosh(a x) + q
+--R /
+--R +---------+
+--R | 2 2
+--R a\|- q + p
+--R ,
+--R +-------+
+--R | 2 2
+--R (q sinh(a x) + q cosh(a x) + p)\|q - p
+--R 2atan(-----------------------------------------)
+--R 2 2
+--R q - p
+--R ------------------------------------------------]
+--R +-------+
+--R | 2 2
+--R a\|q - p
+--R Type: Union(List Expression
Integer,...)
+--E
+
+--S 131
+bb1:=(q*sinh(a*x))/(a*(q^2-p^2)*(p+q*cosh(a*x)))-p/(q^2-p^2)*t1.1
+--R
+--R (3)
+--R 2
+--R (- p q cosh(a x) - p )
+--R *
+--R log
+--R 2 2 2 2
2
+--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q
cosh(a x)
+--R +
+--R 2 2
+--R 2p q cosh(a x) - q + 2p
+--R *
+--R +---------+
+--R | 2 2
+--R \|- q + p
+--R +
+--R 3 2 3 2 2 3
+--R (2q - 2p q)sinh(a x) + (2q - 2p q)cosh(a x) + 2p q - 2p
+--R /
+--R 2 2
+--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x)
+--R +
+--R 2p cosh(a x) + q
+--R +
+--R +---------+
+--R | 2 2
+--R q sinh(a x)\|- q + p
+--R /
+--R +---------+
+--R 3 2 2 3 | 2 2
+--R ((a q - a p q)cosh(a x) + a p q - a p )\|- q + p
+--R Type: Expression
Integer
+--E
+
+--S 132
+bb2:=(q*sinh(a*x))/(a*(q^2-p^2)*(p+q*cosh(a*x)))-p/(q^2-p^2)*t1.2
+--R
+--R (4)
+--R
+-------+
+--R | 2 2
+--R 2 (q sinh(a x) + q cosh(a x) + p)\|q - p
+--R (- 2p q cosh(a x) - 2p
)atan(-----------------------------------------)
+--R 2 2
+--R q - p
+--R +
+--R +-------+
+--R | 2 2
+--R q sinh(a x)\|q - p
+--R /
+--R +-------+
+--R 3 2 2 3 | 2 2
+--R ((a q - a p q)cosh(a x) + a p q - a p )\|q - p
+--R Type: Expression
Integer
+--E
+
+--S 133
+cc1:=aa.1-bb1
+--R
+--R (5)
+--R 2 2 2
+--R (p q cosh(a x) + p q)sinh(a x)
+--R +
+--R 2 2 2 3 2
3
+--R (2p q cosh(a x) + 4p q cosh(a x) + 2p )sinh(a x) + p q cosh(a x)
+--R +
+--R 2 2 2 3 2
+--R 3p q cosh(a x) + (p q + 2p )cosh(a x) + p q
+--R *
+--R log
+--R 2 2 2 2
2
+--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q
cosh(a x)
+--R +
+--R 2 2
+--R 2p q cosh(a x) - q + 2p
+--R *
+--R +---------+
+--R | 2 2
+--R \|- q + p
+--R +
+--R 3 2 3 2 2 3
+--R (2q - 2p q)sinh(a x) + (2q - 2p q)cosh(a x) + 2p q - 2p
+--R /
+--R 2 2
+--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x)
+--R +
+--R 2p cosh(a x) + q
+--R +
+--R 2 2 2
+--R (p q cosh(a x) + p q)sinh(a x)
+--R +
+--R 2 2 2 3 2
3
+--R (2p q cosh(a x) + 4p q cosh(a x) + 2p )sinh(a x) + p q cosh(a x)
+--R +
+--R 2 2 2 3 2
+--R 3p q cosh(a x) + (p q + 2p )cosh(a x) + p q
+--R *
+--R log
+--R 2 2 2 2
2
+--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q
cosh(a x)
+--R +
+--R 2 2
+--R 2p q cosh(a x) - q + 2p
+--R *
+--R +---------+
+--R | 2 2
+--R \|- q + p
+--R +
+--R 3 2 3 2 2
3
+--R (- 2q + 2p q)sinh(a x) + (- 2q + 2p q)cosh(a x) - 2p q +
2p
+--R /
+--R 2 2
+--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x)
+--R +
+--R 2p cosh(a x) + q
+--R +
+--R 2 3 2 2
+--R - q sinh(a x) + (- 2q cosh(a x) - 2p q)sinh(a x)
+--R +
+--R 2 2 2 2
+--R (- q cosh(a x) - 4p q cosh(a x) - q - 2p )sinh(a x)
+--R +
+--R 2 2 2
+--R - 2p q cosh(a x) + (- 2q - 2p )cosh(a x) - 2p q
+--R *
+--R +---------+
+--R | 2 2
+--R \|- q + p
+--R /
+--R 4 2 2 3 3 2
+--R ((a q - a p q )cosh(a x) + a p q - a p q)sinh(a x)
+--R +
+--R 4 2 2 2 3 3
2 2
+--R (2a q - 2a p q )cosh(a x) + (4a p q - 4a p q)cosh(a x) + 2a
p q
+--R +
+--R 4
+--R - 2a p
+--R *
+--R sinh(a x)
+--R +
+--R 4 2 2 3 3 3 2
+--R (a q - a p q )cosh(a x) + (3a p q - 3a p q)cosh(a x)
+--R +
+--R 4 2 2 4 3 3
+--R (a q + a p q - 2a p )cosh(a x) + a p q - a p q
+--R *
+--R +---------+
+--R | 2 2
+--R \|- q + p
+--R Type: Expression
Integer
+--E
+
+--S 134
+cc2:=aa.2-bb1
+--R
+--R (6)
+--R 2 2 2
+--R (p q cosh(a x) + p q)sinh(a x)
+--R +
+--R 2 2 2 3 2
3
+--R (2p q cosh(a x) + 4p q cosh(a x) + 2p )sinh(a x) + p q cosh(a x)
+--R +
+--R 2 2 2 3 2
+--R 3p q cosh(a x) + (p q + 2p )cosh(a x) + p q
+--R *
+--R +-------+
+--R | 2 2
+--R \|q - p
+--R *
+--R log
+--R 2 2 2 2
2
+--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q
cosh(a x)
+--R +
+--R 2 2
+--R 2p q cosh(a x) - q + 2p
+--R *
+--R +---------+
+--R | 2 2
+--R \|- q + p
+--R +
+--R 3 2 3 2 2 3
+--R (2q - 2p q)sinh(a x) + (2q - 2p q)cosh(a x) + 2p q - 2p
+--R /
+--R 2 2
+--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x)
+--R +
+--R 2p cosh(a x) + q
+--R +
+--R 2 2 2
+--R (- 2p q cosh(a x) - 2p q)sinh(a x)
+--R +
+--R 2 2 2 3 2
3
+--R (- 4p q cosh(a x) - 8p q cosh(a x) - 4p )sinh(a x) - 2p q
cosh(a x)
+--R +
+--R 2 2 2 3 2
+--R - 6p q cosh(a x) + (- 2p q - 4p )cosh(a x) - 2p q
+--R *
+--R +-------+
+--R +---------+ | 2 2
+--R | 2 2 (q sinh(a x) + q cosh(a x) + p)\|q - p
+--R \|- q + p atan(-----------------------------------------)
+--R 2 2
+--R q - p
+--R +
+--R 2 3 2 2
+--R - q sinh(a x) + (- 2q cosh(a x) - 2p q)sinh(a x)
+--R +
+--R 2 2 2 2
+--R (- q cosh(a x) - 4p q cosh(a x) - q - 2p )sinh(a x)
+--R +
+--R 2 2 2
+--R - 2p q cosh(a x) + (- 2q - 2p )cosh(a x) - 2p q
+--R *
+--R +---------+ +-------+
+--R | 2 2 | 2 2
+--R \|- q + p \|q - p
+--R /
+--R 4 2 2 3 3 2
+--R ((a q - a p q )cosh(a x) + a p q - a p q)sinh(a x)
+--R +
+--R 4 2 2 2 3 3
2 2
+--R (2a q - 2a p q )cosh(a x) + (4a p q - 4a p q)cosh(a x) + 2a
p q
+--R +
+--R 4
+--R - 2a p
+--R *
+--R sinh(a x)
+--R +
+--R 4 2 2 3 3 3 2
+--R (a q - a p q )cosh(a x) + (3a p q - 3a p q)cosh(a x)
+--R +
+--R 4 2 2 4 3 3
+--R (a q + a p q - 2a p )cosh(a x) + a p q - a p q
+--R *
+--R +---------+ +-------+
+--R | 2 2 | 2 2
+--R \|- q + p \|q - p
+--R Type: Expression
Integer
+--E
+
+--S 135
+cc3:=aa.1-bb2
+--R
+--R (7)
+--R 2 2 2
+--R (p q cosh(a x) + p q)sinh(a x)
+--R +
+--R 2 2 2 3 2
3
+--R (2p q cosh(a x) + 4p q cosh(a x) + 2p )sinh(a x) + p q cosh(a x)
+--R +
+--R 2 2 2 3 2
+--R 3p q cosh(a x) + (p q + 2p )cosh(a x) + p q
+--R *
+--R +-------+
+--R | 2 2
+--R \|q - p
+--R *
+--R log
+--R 2 2 2 2
2
+--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q
cosh(a x)
+--R +
+--R 2 2
+--R 2p q cosh(a x) - q + 2p
+--R *
+--R +---------+
+--R | 2 2
+--R \|- q + p
+--R +
+--R 3 2 3 2 2
3
+--R (- 2q + 2p q)sinh(a x) + (- 2q + 2p q)cosh(a x) - 2p q +
2p
+--R /
+--R 2 2
+--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x)
+--R +
+--R 2p cosh(a x) + q
+--R +
+--R 2 2 2
+--R (2p q cosh(a x) + 2p q)sinh(a x)
+--R +
+--R 2 2 2 3 2
3
+--R (4p q cosh(a x) + 8p q cosh(a x) + 4p )sinh(a x) + 2p q cosh(a
x)
+--R +
+--R 2 2 2 3 2
+--R 6p q cosh(a x) + (2p q + 4p )cosh(a x) + 2p q
+--R *
+--R +-------+
+--R +---------+ | 2 2
+--R | 2 2 (q sinh(a x) + q cosh(a x) + p)\|q - p
+--R \|- q + p atan(-----------------------------------------)
+--R 2 2
+--R q - p
+--R +
+--R 2 3 2 2
+--R - q sinh(a x) + (- 2q cosh(a x) - 2p q)sinh(a x)
+--R +
+--R 2 2 2 2
+--R (- q cosh(a x) - 4p q cosh(a x) - q - 2p )sinh(a x)
+--R +
+--R 2 2 2
+--R - 2p q cosh(a x) + (- 2q - 2p )cosh(a x) - 2p q
+--R *
+--R +---------+ +-------+
+--R | 2 2 | 2 2
+--R \|- q + p \|q - p
+--R /
+--R 4 2 2 3 3 2
+--R ((a q - a p q )cosh(a x) + a p q - a p q)sinh(a x)
+--R +
+--R 4 2 2 2 3 3
2 2
+--R (2a q - 2a p q )cosh(a x) + (4a p q - 4a p q)cosh(a x) + 2a
p q
+--R +
+--R 4
+--R - 2a p
+--R *
+--R sinh(a x)
+--R +
+--R 4 2 2 3 3 3 2
+--R (a q - a p q )cosh(a x) + (3a p q - 3a p q)cosh(a x)
+--R +
+--R 4 2 2 4 3 3
+--R (a q + a p q - 2a p )cosh(a x) + a p q - a p q
+--R *
+--R +---------+ +-------+
+--R | 2 2 | 2 2
+--R \|- q + p \|q - p
+--R Type: Expression
Integer
+--E
+
+--S 136 14:582 Axiom cannot simplify this expression
+cc4:=aa.2-bb2
+--R
+--R (8)
+--R 2 3 2 2
+--R - q sinh(a x) + (- 2q cosh(a x) - 2p q)sinh(a x)
+--R +
+--R 2 2 2 2
2
+--R (- q cosh(a x) - 4p q cosh(a x) - q - 2p )sinh(a x) - 2p q cosh(a
x)
+--R +
+--R 2 2
+--R (- 2q - 2p )cosh(a x) - 2p q
+--R /
+--R 4 2 2 3 3 2
+--R ((a q - a p q )cosh(a x) + a p q - a p q)sinh(a x)
+--R +
+--R 4 2 2 2 3 3
2 2
+--R (2a q - 2a p q )cosh(a x) + (4a p q - 4a p q)cosh(a x) + 2a p
q
+--R +
+--R 4
+--R - 2a p
+--R *
+--R sinh(a x)
+--R +
+--R 4 2 2 3 3 3 2
+--R (a q - a p q )cosh(a x) + (3a p q - 3a p q)cosh(a x)
+--R +
+--R 4 2 2 4 3 3
+--R (a q + a p q - 2a p )cosh(a x) + a p q - a p q
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.583~~~~~$\displaystyle
@@ -594,7 +2409,7 @@ $$
<<*>>=
)clear all
---S 22 of 28
+--S 137
aa:=integrate(1/(p^2-q^2*cosh(a*x)^2),x)
--R
--R
@@ -661,6 +2476,219 @@ aa:=integrate(1/(p^2-q^2*cosh(a*x)^2),x)
--R ]
--R Type: Union(List Expression
Integer,...)
--E
+
+--S 138
+bb1:=1/(2*a*p*sqrt(p^2-q^2))*log((p*tanh(a*x)+sqrt(p^2-q^2))/(p*tanh(a*x)-sqrt(p^2-q^2)))
+--R
+--R +---------+
+--R | 2 2
+--R - \|- q + p - p tanh(a x)
+--R log(----------------------------)
+--R +---------+
+--R | 2 2
+--R \|- q + p - p tanh(a x)
+--R (2) ---------------------------------
+--R +---------+
+--R | 2 2
+--R 2a p\|- q + p
+--R Type: Expression
Integer
+--E
+
+--S 139
+bb2:=-1/(a*p*sqrt(q^2-p^2))*atan((p*tanh(a*x))/sqrt(q^2-p^2))
+--R
+--R p tanh(a x)
+--R atan(-----------)
+--R +-------+
+--R | 2 2
+--R \|q - p
+--R (3) - -----------------
+--R +-------+
+--R | 2 2
+--R a p\|q - p
+--R Type: Expression
Integer
+--E
+
+--S 140
+cc1:=aa.1-bb1
+--R
+--R (4)
+--R log
+--R 4 4 4 3
+--R q sinh(a x) + 4q cosh(a x)sinh(a x)
+--R +
+--R 4 2 4 2 2 2
+--R (6q cosh(a x) + 2q - 4p q )sinh(a x)
+--R +
+--R 4 3 4 2 2
+--R (4q cosh(a x) + (4q - 8p q )cosh(a x))sinh(a x)
+--R +
+--R 4 4 4 2 2 2 4 2 2 4
+--R q cosh(a x) + (2q - 4p q )cosh(a x) + q - 8p q + 8p
+--R *
+--R +---------+
+--R | 2 2
+--R \|- q + p
+--R +
+--R 4 3 2 2 4 3 2
+--R (- 4p q + 4p q )sinh(a x) + (- 8p q + 8p q )cosh(a
x)sinh(a x)
+--R +
+--R 4 3 2 2 4 3 2 5
+--R (- 4p q + 4p q )cosh(a x) - 4p q + 12p q - 8p
+--R /
+--R 2 4 2 3
+--R q sinh(a x) + 4q cosh(a x)sinh(a x)
+--R +
+--R 2 2 2 2 2
+--R (6q cosh(a x) + 2q - 4p )sinh(a x)
+--R +
+--R 2 3 2 2 2 4
+--R (4q cosh(a x) + (4q - 8p )cosh(a x))sinh(a x) + q cosh(a x)
+--R +
+--R 2 2 2 2
+--R (2q - 4p )cosh(a x) + q
+--R +
+--R +---------+
+--R | 2 2
+--R - \|- q + p - p tanh(a x)
+--R - log(----------------------------)
+--R +---------+
+--R | 2 2
+--R \|- q + p - p tanh(a x)
+--R /
+--R +---------+
+--R | 2 2
+--R 2a p\|- q + p
+--R Type: Expression
Integer
+--E
+
+--S 141
+cc2:=aa.2-bb1
+--R
+--R (5)
+--R +---------+
+--R +-------+ | 2 2
+--R | 2 2 - \|- q + p - p tanh(a x)
+--R - \|q - p log(----------------------------)
+--R +---------+
+--R | 2 2
+--R \|- q + p - p tanh(a x)
+--R +
+--R -
+--R +---------+
+--R | 2 2
+--R 2\|- q + p
+--R *
+--R atan
+--R 2 2 2 2 2
2
+--R q sinh(a x) + 2q cosh(a x)sinh(a x) + q cosh(a x) + q
+--R +
+--R 2
+--R - 2p
+--R *
+--R +-------+
+--R | 2 2
+--R \|q - p
+--R /
+--R 2 3
+--R 2p q - 2p
+--R /
+--R +---------+ +-------+
+--R | 2 2 | 2 2
+--R 2a p\|- q + p \|q - p
+--R Type: Expression
Integer
+--E
+
+--S 142
+cc3:=aa.1-bb2
+--R
+--R (6)
+--R +-------+
+--R | 2 2
+--R \|q - p
+--R *
+--R log
+--R 4 4 4 3
+--R q sinh(a x) + 4q cosh(a x)sinh(a x)
+--R +
+--R 4 2 4 2 2 2
+--R (6q cosh(a x) + 2q - 4p q )sinh(a x)
+--R +
+--R 4 3 4 2 2
+--R (4q cosh(a x) + (4q - 8p q )cosh(a x))sinh(a x)
+--R +
+--R 4 4 4 2 2 2 4 2 2
4
+--R q cosh(a x) + (2q - 4p q )cosh(a x) + q - 8p q + 8p
+--R *
+--R +---------+
+--R | 2 2
+--R \|- q + p
+--R +
+--R 4 3 2 2
+--R (- 4p q + 4p q )sinh(a x)
+--R +
+--R 4 3 2
+--R (- 8p q + 8p q )cosh(a x)sinh(a x)
+--R +
+--R 4 3 2 2 4 3 2 5
+--R (- 4p q + 4p q )cosh(a x) - 4p q + 12p q - 8p
+--R /
+--R 2 4 2 3
+--R q sinh(a x) + 4q cosh(a x)sinh(a x)
+--R +
+--R 2 2 2 2 2
+--R (6q cosh(a x) + 2q - 4p )sinh(a x)
+--R +
+--R 2 3 2 2 2
4
+--R (4q cosh(a x) + (4q - 8p )cosh(a x))sinh(a x) + q cosh(a
x)
+--R +
+--R 2 2 2 2
+--R (2q - 4p )cosh(a x) + q
+--R +
+--R +---------+
+--R | 2 2 p tanh(a x)
+--R 2\|- q + p atan(-----------)
+--R +-------+
+--R | 2 2
+--R \|q - p
+--R /
+--R +---------+ +-------+
+--R | 2 2 | 2 2
+--R 2a p\|- q + p \|q - p
+--R Type: Expression
Integer
+--E
+
+--S 143 14:583 Axiom cannot simplify this expression
+cc4:=aa.2-bb2
+--R
+--R (7)
+--R -
+--R atan
+--R 2 2 2 2 2
2
+--R q sinh(a x) + 2q cosh(a x)sinh(a x) + q cosh(a x) + q
+--R +
+--R 2
+--R - 2p
+--R *
+--R +-------+
+--R | 2 2
+--R \|q - p
+--R /
+--R 2 3
+--R 2p q - 2p
+--R +
+--R p tanh(a x)
+--R atan(-----------)
+--R +-------+
+--R | 2 2
+--R \|q - p
+--R /
+--R +-------+
+--R | 2 2
+--R a p\|q - p
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.584~~~~~$\displaystyle
@@ -680,7 +2708,7 @@ $$
<<*>>=
)clear all
---S 23 of 28
+--S 144
aa:=integrate(1/(p^2+q^2*cosh(a*x)^2),x)
--R
--R
@@ -725,6 +2753,142 @@ aa:=integrate(1/(p^2+q^2*cosh(a*x)^2),x)
--R 2a p\|q + p
--R Type: Union(Expression
Integer,...)
--E
+
+--S 145
+bb1:=1/(2*a*p*sqrt(p^2+q^2))*log((p*tanh(a*x)+sqrt(p^2+q^2))/(p*tanh(a*x)-sqrt(p^2+q^2)))
+--R
+--R +-------+
+--R | 2 2
+--R - \|q + p - p tanh(a x)
+--R log(--------------------------)
+--R +-------+
+--R | 2 2
+--R \|q + p - p tanh(a x)
+--R (2) -------------------------------
+--R +-------+
+--R | 2 2
+--R 2a p\|q + p
+--R Type: Expression
Integer
+--E
+
+--S 146
+bb2:=1/(a*p*sqrt(p^2+q^2))*atan((p*tanh(a*x))/sqrt(p^2+q^2))
+--R
+--R p tanh(a x)
+--R atan(-----------)
+--R +-------+
+--R | 2 2
+--R \|q + p
+--R (3) -----------------
+--R +-------+
+--R | 2 2
+--R a p\|q + p
+--R Type: Expression
Integer
+--E
+
+--S 147
+cc1:=aa-bb1
+--R
+--R (4)
+--R log
+--R 4 4 4 3
+--R q sinh(a x) + 4q cosh(a x)sinh(a x)
+--R +
+--R 4 2 4 2 2 2
+--R (6q cosh(a x) + 2q + 4p q )sinh(a x)
+--R +
+--R 4 3 4 2 2
+--R (4q cosh(a x) + (4q + 8p q )cosh(a x))sinh(a x)
+--R +
+--R 4 4 4 2 2 2 4 2 2 4
+--R q cosh(a x) + (2q + 4p q )cosh(a x) + q + 8p q + 8p
+--R *
+--R +-------+
+--R | 2 2
+--R \|q + p
+--R +
+--R 4 3 2 2 4 3 2
+--R (- 4p q - 4p q )sinh(a x) + (- 8p q - 8p q )cosh(a
x)sinh(a x)
+--R +
+--R 4 3 2 2 4 3 2 5
+--R (- 4p q - 4p q )cosh(a x) - 4p q - 12p q - 8p
+--R /
+--R 2 4 2 3
+--R q sinh(a x) + 4q cosh(a x)sinh(a x)
+--R +
+--R 2 2 2 2 2
+--R (6q cosh(a x) + 2q + 4p )sinh(a x)
+--R +
+--R 2 3 2 2 2 4
+--R (4q cosh(a x) + (4q + 8p )cosh(a x))sinh(a x) + q cosh(a x)
+--R +
+--R 2 2 2 2
+--R (2q + 4p )cosh(a x) + q
+--R +
+--R +-------+
+--R | 2 2
+--R - \|q + p - p tanh(a x)
+--R - log(--------------------------)
+--R +-------+
+--R | 2 2
+--R \|q + p - p tanh(a x)
+--R /
+--R +-------+
+--R | 2 2
+--R 2a p\|q + p
+--R Type: Expression
Integer
+--E
+
+--S 148 14:584 Axiom cannot compute this integral
+cc2:=aa-bb2
+--R
+--R (5)
+--R log
+--R 4 4 4 3
+--R q sinh(a x) + 4q cosh(a x)sinh(a x)
+--R +
+--R 4 2 4 2 2 2
+--R (6q cosh(a x) + 2q + 4p q )sinh(a x)
+--R +
+--R 4 3 4 2 2
+--R (4q cosh(a x) + (4q + 8p q )cosh(a x))sinh(a x)
+--R +
+--R 4 4 4 2 2 2 4 2 2 4
+--R q cosh(a x) + (2q + 4p q )cosh(a x) + q + 8p q + 8p
+--R *
+--R +-------+
+--R | 2 2
+--R \|q + p
+--R +
+--R 4 3 2 2 4 3 2
+--R (- 4p q - 4p q )sinh(a x) + (- 8p q - 8p q )cosh(a
x)sinh(a x)
+--R +
+--R 4 3 2 2 4 3 2 5
+--R (- 4p q - 4p q )cosh(a x) - 4p q - 12p q - 8p
+--R /
+--R 2 4 2 3
+--R q sinh(a x) + 4q cosh(a x)sinh(a x)
+--R +
+--R 2 2 2 2 2
+--R (6q cosh(a x) + 2q + 4p )sinh(a x)
+--R +
+--R 2 3 2 2 2 4
+--R (4q cosh(a x) + (4q + 8p )cosh(a x))sinh(a x) + q cosh(a x)
+--R +
+--R 2 2 2 2
+--R (2q + 4p )cosh(a x) + q
+--R +
+--R p tanh(a x)
+--R - 2atan(-----------)
+--R +-------+
+--R | 2 2
+--R \|q + p
+--R /
+--R +-------+
+--R | 2 2
+--R 2a p\|q + p
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.585~~~~~$\displaystyle
@@ -735,7 +2899,7 @@ $$
<<*>>=
)clear all
---S 24 of 28
+--S 149 14:585 Axiom cannot compute this integral
aa:=integrate(x^m*cosh(a*x),x)
--R
--R
@@ -755,7 +2919,7 @@ $$
<<*>>=
)clear all
---S 25 of 28
+--S 150 14:586 Axiom cannot compute this integral
aa:=integrate(cosh(a*x)^n,x)
--R
--R
@@ -776,7 +2940,7 @@ $$
<<*>>=
)clear all
---S 26 of 28
+--S 151 14:587 Axiom cannot compute this integral
aa:=integrate(cosh(a*x)/x^n,x)
--R
--R
@@ -798,7 +2962,7 @@ $$
<<*>>=
)clear all
---S 27 of 28
+--S 152 14:588 Axiom cannot compute this integral
aa:=integrate(1/cosh(a*x)^n,x)
--R
--R
@@ -821,7 +2985,7 @@ $$
<<*>>=
)clear all
---S 28 of 28
+--S 153 14:589 Axiom cannot compute this integral
aa:=integrate(1/cosh(a*x)^n,x)
--R
--R
- [Axiom-developer] 20080426.01.tpd.patch (CATS Schaums-Axiom equivalence testing (26-28)),
daly <=