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[Axiom-developer] 20080418.01.tpd.patch (CATS Schaums-Axiom equivalence
|
From: |
daly |
|
Subject: |
[Axiom-developer] 20080418.01.tpd.patch (CATS Schaums-Axiom equivalence testing (1-11)) |
|
Date: |
Sat, 19 Apr 2008 01:28:16 -0500 |
This patch revises several old files where it was claimed that Axiom
could not factor the results. As Shakespeare once said (but never
wrote down), "The fault, Dear Brutus, is not in our software but
in ourselves". Now that I'm a tad wiser about simplification in Axiom
I was able to simplify most of the results.
Also of interest, I believe that there is a typo in the Schaums book
for problem 14:91. Axiom cannot simplify this expression. It appears
that the trailing sqrt factor should be
sqrt((a*x+b)^3)
not
sqrt((a+b*x)^3)
For more information look at schaum2.input.pamphlet,--S 56 where I
factor the numerator of the Axiom result. Note that I'm using the
Schaums book published in 1968.
Other than 14:91, where Axiom could compute the integral, the results
to date have agreed with Schaums to within a constant of integration.
schaum11.input.pamphlet is new work. The other changes are rework
of prior results.
The new cumulative results are:
schaum1.input.pamphet
14:59 Schaums and Axiom agree
14:60 Schaums and Axiom agree
14:61 Schaums and Axiom differ by a constant
14:62 Schaums and Axiom differ by a constant
14:63 Schaums and Axiom agree
14:64 Schaums and Axiom agree
14:65 Schaums and Axiom agree
14:66 Schaums and Axiom agree
14:67 Schaums and Axiom agree
14:68 Schaums and Axiom differ by a constant
14:69 Schaums and Axiom differ by a constant
14:70 Schaums and Axiom agree
14:71 Schaums and Axiom agree
14:72 Schaums and Axiom differ by a constant
14:73 Schaums and Axiom differ by a constant
14:74 Schaums and Axiom agree
14:75 Schaums and Axiom agree
14:76 Schaums and Axiom agree
14:77 Schaums and Axiom differ by a constant
14:78 Schaums and Axiom agree
14:79 Schaums and Axiom agree
14:80 Schaums and Axiom agree
14:82 Schaums and Axiom agree
14:83 Axiom cannot do this integration
schaum2.input.pamphet
14:84 Schaums and Axiom agree
14:85 Schaums and Axiom agree
14:86 Schaums and Axiom agree
14:87a Schaums and Axiom differ by a constant
14:87b Schaums and Axiom differ by a constant
14:88 Schaums and Axiom differ by a constant
14:89 Schaums and Axiom differ by a constant
14:90 Schaums and Axiom agree
14:91 Axiom cannot simplify this expression. Schaums typo?
14:92 Axiom cannot simplify this expression
14:93 Schaums and Axiom agree
14:94 Axiom cannot do this integral
14:95 Axiom cannot do this integral
14:96 Axiom cannot do this integral
14:97 Axiom cannot do this integral
14:98 Axiom cannot do this integral
14:99 Schaums and Axiom agree
14:100 Schaums and Axiom agree
14:101 Schaums and Axiom agree
14:102 Axiom cannot do this integral
14:103 Axiom cannot do this integral
14:104 Axiom cannot do this integral
schaum3.input.pamphet
14:105 Schaums and Axiom agree
14:106 Schaums and Axiom agree
14:107 Schaums and Axiom agree
14:108 Schaums and Axiom agree
14:109 Schaums and Axiom agree
14:110 Axiom cannot do this integral
14:111 Schaums and Axiom agree
14:112 Axiom cannot do this integral
schaum4.input.pamphet
14:113 Schaums and Axiom agree
14:114 Axiom cannot simplify these answers
14:115 Axiom cannot simplify these answers
14:116 Axiom cannot compute this integral
14:117 Axiom cannot compute this integral
14:118 Axiom cannot compute this integral
14:119 Axiom cannot compute this integral
schaum5.input.pamphet
14:120 Axiom cannot simplify these answers
14:121 Axiom cannot simplify this answer
14:122 Axiom cannot simplify this answer
14:123 Axiom cannot simplify these results
14:124 Axiom cannot simplify this result
schaum6.input.pamphet
14:125 Schaums and Axiom agree
14:126 Schaums and Axiom agree
14:127 Schaums and Axiom agree
14:128 Schaums and Axiom agree
14:129 Schaums and Axiom agree
14:130 Schaums and Axiom agree
14:131 Schaums and Axiom agree
14:132 Schaums and Axiom agree
14:133 Schaums and Axiom agree
14:134 Schaums and Axiom differ by a constant
14:135 Schaums and Axiom agree
14:136 Schaums and Axiom agree
14:137 Schaums and Axiom agree
14:138 Schaums and Axiom agree
14:139 Axiom cannot do this integral
14:140 Schaums and Axiom agree
14:141 Axiom cannot do this integral
14:142 Axiom cannot do this integral
14:143 Axiom cannot do this integral
schaum7.input.pamphet
14:144 Schaums and Axiom agree
14:145 Schaums and Axiom agree
14:146 Schaums and Axiom agree
14:147 Schaums and Axiom agree
14:148 Schaums and Axiom agree
14:149 Schaums and Axiom agree
14:150 Schaums and Axiom agree
14:151 Schaums and Axiom agree
14:152 Schaums and Axiom agree
14:153 Schaums and Axiom agree
14:154 Schaums and Axiom agree
14:155 Schaums and Axiom agree
14:156 Schaums and Axiom agree
14:157 Schaums and Axiom agree
14:158 Axiom cannot do this integral
14:159 Schaums and Axiom agree
14:160 Axiom cannot compute this integral
14:161 Axiom cannot compute this integral
14:162 Axiom cannot compute this integral
schaum8.input.pamphet
14:163 Schaums and Axiom differ by a constant
14:164 Schaums and Axiom differ by a constant
14:165 Schaums and Axiom differ by a constant
14:166 Schaums and Axiom differ by a constant
14:167 Schaums and Axiom differ by a constant
14:168 Schaums and Axiom differ by a constant
14:169 Schaums and Axiom differ by a constant
14:170 Schaums and Axiom differ by a constant
14:171 Schaums and Axiom agree
14:172 Schaums and Axiom differ by a constant
14:173 Schaums and Axiom differ by a constant
14:174 Schaums and Axiom differ by a constant
14:175 Schaums and Axiom differ by a constant
14:176 Schaums and Axiom differ by a constant
14:177 Axiom cannot do this integration
14:178 Schaums and Axiom agree
14:179 Axiom cannot integrate this expression
14:180 Axiom cannot integrate this expression
14:181 Axiom cannot integrate this expression
schaum9.input.pamphet
14:182 Schaums and Axiom differ by a constant
14:183 Schaums and Axiom agree
14:184 Schaums and Axiom differ by a constant
14:185 Schaums and Axiom agree
14:186 Schaums and Axiom differ by a constant
14:187 Schaums and Axiom differ by a constant
14:188 Schaums and Axiom differ by a constant
14:189 Schaums and Axiom differ by a constant
14:190 Schaums and Axiom agree
14:191 Schaums and Axiom differ by a constant
14:192 Schaums and Axiom agree
14:193 Schaums and Axiom differ by a constant
14:194 Schaums and Axiom differ by a constant
14:195 Schaums and Axiom differ by a constant
14:196 Schaums and Axiom differ by a constant
14:197 Schaums and Axiom agree
14:198 Schaums and Axiom differ by a constant
14:199 Schaums and Axiom agree
14:200 Schaums and Axiom differ by a constant
14:201 Schaums and Axiom differ by a constant
14:202 Schaums and Axiom differ by a constant
14:203 Schaums and Axiom differ by a constant
14:204 Schaums and Axiom agree
14:205 Schaums and Axiom differ by a constant
14:206 Schaums and Axiom agree
14:207 Schaums and Axiom differ by a constant
14:208 Schaums and Axiom differ by a constant
14:209 Schaums and Axiom differ by a constant
schaum10.input.pamphet
14:210 Schaums and Axiom differ by a constant
14:xxx Schaums and Axiom agree
14:211 Schaums and Axiom differ by a constant
14:212 Schaums and Axiom agree
14:213 Schaums and Axiom differ by a constant
14:214 Schaums and Axiom differ by a constant
14:215 Schaums and Axiom differ by a constant
14:216 Schaums and Axiom differ by a constant
14:217 Schaums and Axiom agree
14:218 Schaums and Axiom differ by a constant
14:219 Schaums and Axiom agree
14:220 Schaums and Axiom differ by a constant
14:221 Schaums and Axiom differ by a constant
14:222 Schaums and Axiom differ by a constant
14:223 Schaums and Axiom differ by a constant
14:224 Schaums and Axiom agree
14:225 Schaums and Axiom differ by a constant
14:226 Schaums and Axiom agree
14:227 Schaums and Axiom differ by a constant
14:228 Schaums and Axiom differ by a constant
14:229 Schaums and Axiom differ by a constant
14:230 Schaums and Axiom differ by a constant
14:231 Schaums and Axiom agree
14:232 Schaums and Axiom differ by a constant
14:233 Schaums and Axiom agree
14:234 Schaums and Axiom differ by a constant
14:235 Schaums and Axiom differ by a constant
14:236 Schaums and Axiom differ by a constant
schaum11.input.pamphet
14:238 Schaums and Axiom agree
14:238 Schaums and Axiom differ by a constant
14:239 Schaums and Axiom agree
14:240 Schaums and Axiom differ by a constant
14:241 Schaums and Axiom differ by a constant
14:242 Schaums and Axiom agree
14:243 Schaums and Axiom differ by a constant
14:244 Schaums and Axiom agree
14:245 Schaums and Axiom differ by a constant
14:246 Schaums and Axiom agree
14:247 Schaums and Axiom differ by a constant
14:248 Schaums and Axiom differ by a constant
14:249 Schaums and Axiom agree
14:250 Schaums and Axiom differ by a constant
14:251 Schaums and Axiom agree
14:252 Schaums and Axiom differ by a constant
14:253 Schaums and Axiom agree
14:254 Schaums and Axiom differ by a constant
14:255 Schaums and Axiom differ by a constant
14:256 Schaums and Axiom agree
14:257 Schaums and Axiom differ by a constant
14:258 Schaums and Axiom agree
14:259 Schaums and Axiom differ by a constant
14:260 Schaums and Axiom agree
14:261 Schaums and Axiom differ by a constant
14:262 Schaums and Axiom differ by a constant
14:263 Schaums and Axiom agree
14:264 Schaums and Axiom differ by a constant
=========================================================================
diff --git a/changelog b/changelog
index 502aa1e..c10550c 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,11 @@
+20080418 tpd src/input/schaum11.input show Schaums-Axiom equivalence
+20080418 tpd src/input/schaum10.input fix Schaums-Axiom equivalence
+20080418 tpd src/input/schaum9.input fix Schaums-Axiom equivalence
+20080418 tpd src/input/schaum8.input fix Schaums-Axiom equivalence
+20080418 tpd src/input/schaum7.input fix Schaums-Axiom equivalence
+20080418 tpd src/input/schaum6.input fix Schaums-Axiom equivalence
+20080418 tpd src/input/schaum2.input fix Schaums-Axiom equivalence
+20080418 tpd src/input/schaum1.input fix Schaums-Axiom equivalence
20080417 tpd src/input/schaum10.input show Schaums-Axiom equivalence
20080417 tpd src/input/schaum9.input show Schaums-Axiom equivalence
20080417 tpd src/input/schaum8.input show Schaums-Axiom equivalence
diff --git a/src/input/schaum1.input.pamphlet b/src/input/schaum1.input.pamphlet
index 7e7e8c4..615d907 100644
--- a/src/input/schaum1.input.pamphlet
+++ b/src/input/schaum1.input.pamphlet
@@ -1201,7 +1201,7 @@ explog:=rule(%e^(n*log(x)) == x^n)
--R Type: RewriteRule(Integer,Integer,Expression
Integer)
--E
---S 94 14:80 Schaums and Axiom agree
+--S 94
dd:=explog cc
--R
--R n + 1 n
@@ -1210,8 +1210,14 @@ dd:=explog cc
--R a n + a
--R Type: Expression
Integer
--E
+
+--S 95 14:80 Schaums and Axiom agree
+ee:=complexNormalize dd
+--R
+--R (6) 0
+--R Type: Expression
Integer
+--E
@
-The numerator is clearly zero but I cannot get Axiom to simplify it.
\section{\cite{1}:14.81~~~~~$\displaystyle
\int{x(ax+b)^n~dx}$}
@@ -1221,7 +1227,7 @@ $$\int{x(ax+b)^n}=
$$
<<*>>=
)clear all
---S 95
+--S 96
aa:=integrate(x*(a*x+b)^n,x)
--R
--R 2 2 2 2 n log(a x + b)
@@ -1232,7 +1238,7 @@ aa:=integrate(x*(a*x+b)^n,x)
--R Type: Union(Expression
Integer,...)
--E
---S 96
+--S 97
bb:=((a*x+b)^(n+2))/((n+2)*a^2)-(b*(a*x+b)^(n+1))/((n+1)*a^2)
--R
--R n + 2 n + 1
@@ -1243,7 +1249,7 @@
bb:=((a*x+b)^(n+2))/((n+2)*a^2)-(b*(a*x+b)^(n+1))/((n+1)*a^2)
--R Type: Expression
Integer
--E
---S 97
+--S 98
cc:=aa-bb
--R
--R (3)
@@ -1258,7 +1264,7 @@ cc:=aa-bb
--R Type: Expression
Integer
--E
---S 98
+--S 99
explog:=rule(%e^(n*log(x)) == x^n)
--R
--R n log(x) n
@@ -1266,7 +1272,7 @@ explog:=rule(%e^(n*log(x)) == x^n)
--R Type: RewriteRule(Integer,Integer,Expression
Integer)
--E
---S 99 14:81 Schaums and Axiom agreement cannot be determined
+--S 100
dd:=explog cc
--R
--R (5)
@@ -1280,6 +1286,13 @@ dd:=explog cc
--R a n + 3a n + 2a
--R Type: Expression
Integer
--E
+
+--S 101
+ee:=complexNormalize dd
+--R
+--R (6) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.82~~~~~$\displaystyle
\int{x^2(ax+b)^n~dx}$}
@@ -1292,7 +1305,7 @@ $$
<<*>>=
)clear all
---S 100
+--S 102
aa:=integrate(x^2*(a*x+b)^n,x)
--R
--R (1)
@@ -1304,7 +1317,7 @@ aa:=integrate(x^2*(a*x+b)^n,x)
--R Type: Union(Expression
Integer,...)
--E
---S 101
+--S 103
bb:=(a*x+b)^(n+3)/((n+3)*a^3)-(2*b*(a*x+b)^(n+2))/((n+2)*a^3)+(b^2*(a*x+b)^(n+1))/((n+1)*a^3)
--R
--R (2)
@@ -1319,7 +1332,7 @@
bb:=(a*x+b)^(n+3)/((n+3)*a^3)-(2*b*(a*x+b)^(n+2))/((n+2)*a^3)+(b^2*(a*x+b)^(n+1)
--R Type: Expression
Integer
--E
---S 102 14:82 Schaums and Axiom agreement cannot be determined
+--S 104
cc:=aa-bb
--R
--R (3)
@@ -1339,6 +1352,39 @@ cc:=aa-bb
--R a n + 6a n + 11a n + 6a
--R Type: Expression
Integer
--E
+
+--S 105
+explog:=rule(%e^(n*log(x)) == x^n)
+--R
+--R n log(x) n
+--R (4) %e == x
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 106
+dd:=explog cc
+--R
+--R (5)
+--R 2 n + 3 2 n + 2
+--R (- n - 3n - 2)(a x + b) + (2b n + 8b n + 6b)(a x + b)
+--R +
+--R 2 2 2 2 n + 1
+--R (- b n - 5b n - 6b )(a x + b)
+--R +
+--R 3 2 3 3 3 2 2 2 2 2 3
n
+--R ((a n + 3a n + 2a )x + (a b n + a b n)x - 2a b n x + 2b )(a x +
b)
+--R /
+--R 3 3 3 2 3 3
+--R a n + 6a n + 11a n + 6a
+--R Type: Expression
Integer
+--E
+
+--S 107 14:82 Schaums and Axiom agree
+ee:=complexNormalize dd
+--R
+--R (6) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.83~~~~~$\displaystyle
\int{x^m(ax+b)^n}~dx$}
@@ -1361,7 +1407,7 @@ $$\int{x^m(ax+b)^n}
$$
<<*>>=
---S 103 14:83 Axiom cannot do this integration
+--S 108 14:83 Axiom cannot do this integration
aa:=integrate(x^m*(a*x+b)^n,x)
--R
--R x
diff --git a/src/input/schaum10.input.pamphlet
b/src/input/schaum10.input.pamphlet
index d0ba7d9..6149cc0 100644
--- a/src/input/schaum10.input.pamphlet
+++ b/src/input/schaum10.input.pamphlet
@@ -133,7 +133,7 @@ bb:=(x*sqrt(x^2-a^2))/2+a^2/2*log(x+sqrt(x^2-a^2))
--R Type: Expression
Integer
--E
---S 11 14:211 Axiom cannot simplify this expression
+--S 11
cc:=aa-bb
--R
--R +-------+ +-------+
@@ -144,6 +144,16 @@ cc:=aa-bb
--R Type: Expression
Integer
--E
+--S 12 14:211 Schaums and Axiom differ by a constant
+dd:=complexNormalize cc
+--R
+--R 2 2
+--R a log(- a )
+--R (4) - -----------
+--R 2
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.212~~~~~$\displaystyle
@@ -154,7 +164,7 @@ $$
<<*>>=
)clear all
---S 12
+--S 13
aa:=integrate(x^3/sqrt(x^2-a^2),x)
--R
--R
@@ -168,7 +178,7 @@ aa:=integrate(x^3/sqrt(x^2-a^2),x)
--R Type: Union(Expression
Integer,...)
--E
---S 13
+--S 14
bb:=(x^2-a^2)^(3/2)/3+a^2*sqrt(x^2-a^2)
--R
--R +-------+
@@ -179,7 +189,7 @@ bb:=(x^2-a^2)^(3/2)/3+a^2*sqrt(x^2-a^2)
--R Type: Expression
Integer
--E
---S 14 14:212 Schaums and Axiom agree
+--S 15 14:212 Schaums and Axiom agree
cc:=aa-bb
--R
--R (3) 0
@@ -194,7 +204,7 @@ $$
<<*>>=
)clear all
---S 15
+--S 16
aa:=integrate(1/(x*sqrt(x^2-a^2)),x)
--R
--R
@@ -208,29 +218,134 @@ aa:=integrate(1/(x*sqrt(x^2-a^2)),x)
--R Type: Union(Expression
Integer,...)
--E
---S 16
-bb:=1/a*asec(abs(x/a))
+--S 17
+bb:=1/a*asec(x/a)
--R
---R x
---R asec(abs(-))
---R a
---R (2) ------------
---R a
+--R x
+--R asec(-)
+--R a
+--R (2) -------
+--R a
--R Type: Expression
Integer
--E
---S 17 14:213 Axiom cannot simplify this expression
+--S 18
cc:=aa-bb
--R
--R +-------+
--R | 2 2
---R \|x - a - x x
---R 2atan(--------------) - asec(abs(-))
---R a a
---R (3) ------------------------------------
---R a
+--R \|x - a - x x
+--R 2atan(--------------) - asec(-)
+--R a a
+--R (3) -------------------------------
+--R a
--R Type: Expression
Integer
--E
+
+--S 19
+asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x))
+--R
+--R +------+
+--R | 2
+--R |x - 1
+--R x |------ + %i
+--R | 2
+--R \| x
+--R 2%i log(---------------) + %pi
+--R x
+--R (4) asec(x) == ------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 20
+dd:=asecrule cc
+--R
+--R +-------+
+--R | 2 2
+--R |x - a
+--R x |------- + %i a +-------+
+--R | 2 | 2 2
+--R \| x \|x - a - x
+--R - 2%i log(------------------) + 4atan(--------------) - %pi
+--R x a
+--R (5) -----------------------------------------------------------
+--R 2a
+--R Type: Expression Complex
Integer
+--E
+
+--S 21
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R - x + %i
+--R %i log(--------)
+--R x + %i
+--R (6) atan(x) == - ----------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 22
+ee:=atanrule dd
+--R
+--R +-------+
+--R | 2 2
+--R |x - a
+--R x |------- + %i a +-------+
+--R | 2 | 2 2
+--R \| x - \|x - a + x + %i a
+--R - 2%i log(------------------) - 2%i log(-----------------------) -
%pi
+--R x +-------+
+--R | 2 2
+--R \|x - a - x + %i a
+--R (7)
----------------------------------------------------------------------
+--R 2a
+--R Type: Expression Complex
Integer
+--E
+
+--S 23
+ff:=expandLog ee
+--R
+--R (8)
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R 2%i log(\|x - a - x + %i a) - 2%i log(\|x - a - x - %i a)
+--R +
+--R +-------+
+--R | 2 2
+--R |x - a
+--R - 2%i log(x |------- + %i a) + 2%i log(x) - 2%i log(- 1) - %pi
+--R | 2
+--R \| x
+--R /
+--R 2a
+--R Type: Expression Complex
Integer
+--E
+
+--S 24
+gg:=rootSimp ff
+--R
+--R (9)
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R - 2%i log(\|x - a + %i a) + 2%i log(\|x - a - x + %i a)
+--R +
+--R +-------+
+--R | 2 2
+--R - 2%i log(\|x - a - x - %i a) + 2%i log(x) - 2%i log(- 1) - %pi
+--R /
+--R 2a
+--R Type: Expression Complex
Integer
+--E
+
+--S 25 14:213 Schaums and Axiom differ by a constant
+hh:=complexNormalize gg
+--R
+--R %pi
+--R (10) - ---
+--R 2a
+--R Type: Expression Complex
Integer
+--E
@
\section{\cite{1}:14.214~~~~~$\displaystyle
@@ -241,7 +356,7 @@ $$
<<*>>=
)clear all
---S 18
+--S 26
aa:=integrate(1/(x^2*sqrt(x^2-a^2)),x)
--R
--R
@@ -253,7 +368,7 @@ aa:=integrate(1/(x^2*sqrt(x^2-a^2)),x)
--R Type: Union(Expression
Integer,...)
--E
---S 19
+--S 27
bb:=sqrt(x^2-a^2)/(a^2*x)
--R
--R +-------+
@@ -265,7 +380,7 @@ bb:=sqrt(x^2-a^2)/(a^2*x)
--R Type: Expression
Integer
--E
---S 20 14:214 Schaums and Axiom differ by a constant
+--S 28 14:214 Schaums and Axiom differ by a constant
cc:=aa-bb
--R
--R 1
@@ -284,7 +399,7 @@ $$
<<*>>=
)clear all
---S 21
+--S 29
aa:=integrate(1/(x^3*sqrt(x^2-a^2)),x)
--R
--R
@@ -305,33 +420,143 @@ aa:=integrate(1/(x^3*sqrt(x^2-a^2)),x)
--R Type: Union(Expression
Integer,...)
--E
---S 22
-bb:=sqrt(x^2-a^2)/(2*a^2*x^2)+1/(2*a^3)*asec(abs(x/a))
+--S 30
+bb:=sqrt(x^2-a^2)/(2*a^2*x^2)+1/(2*a^3)*asec(x/a)
--R
---R +-------+
---R 2 x | 2 2
---R x asec(abs(-)) + a\|x - a
---R a
---R (2) ----------------------------
---R 3 2
---R 2a x
+--R +-------+
+--R | 2 2 2 x
+--R a\|x - a + x asec(-)
+--R a
+--R (2) -----------------------
+--R 3 2
+--R 2a x
--R Type: Expression
Integer
--E
---S 23 14:215 Axiom cannot simplify this expression
+--S 31
cc:=aa-bb
+--R
--R
--R +-------+
--R | 2 2
---R \|x - a - x x
---R 2atan(--------------) - asec(abs(-))
---R a a
---R (3) ------------------------------------
---R 3
---R 2a
+--R \|x - a - x x
+--R 2atan(--------------) - asec(-)
+--R a a
+--R (3) -------------------------------
+--R 3
+--R 2a
--R Type: Expression
Integer
--E
+--S 32
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R - x + %i
+--R %i log(--------)
+--R x + %i
+--R (4) atan(x) == - ----------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 33
+dd:=atanrule cc
+--R
+--R +-------+
+--R | 2 2
+--R - \|x - a + x + %i a x
+--R - %i log(-----------------------) - asec(-)
+--R +-------+ a
+--R | 2 2
+--R \|x - a - x + %i a
+--R (5) -------------------------------------------
+--R 3
+--R 2a
+--R Type: Expression Complex
Integer
+--E
+
+--S 34
+asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x))
+--R
+--R +------+
+--R | 2
+--R |x - 1
+--R x |------ + %i
+--R | 2
+--R \| x
+--R 2%i log(---------------) + %pi
+--R x
+--R (6) asec(x) == ------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 35
+ee:=asecrule dd
+--R
+--R +-------+
+--R | 2 2
+--R |x - a
+--R x |------- + %i a +-------+
+--R | 2 | 2 2
+--R \| x - \|x - a + x + %i a
+--R - 2%i log(------------------) - 2%i log(-----------------------) -
%pi
+--R x +-------+
+--R | 2 2
+--R \|x - a - x + %i a
+--R (7)
----------------------------------------------------------------------
+--R 3
+--R 4a
+--R Type: Expression Complex
Integer
+--E
+
+--S 36
+ff:=expandLog ee
+--R
+--R (8)
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R 2%i log(\|x - a - x + %i a) - 2%i log(\|x - a - x - %i a)
+--R +
+--R +-------+
+--R | 2 2
+--R |x - a
+--R - 2%i log(x |------- + %i a) + 2%i log(x) - 2%i log(- 1) - %pi
+--R | 2
+--R \| x
+--R /
+--R 3
+--R 4a
+--R Type: Expression Complex
Integer
+--E
+
+--S 37
+gg:=rootSimp ff
+--R
+--R (9)
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R - 2%i log(\|x - a + %i a) + 2%i log(\|x - a - x + %i a)
+--R +
+--R +-------+
+--R | 2 2
+--R - 2%i log(\|x - a - x - %i a) + 2%i log(x) - 2%i log(- 1) - %pi
+--R /
+--R 3
+--R 4a
+--R Type: Expression Complex
Integer
+--E
+
+--S 38 14:215 Schaums and Axiom differ by a constant
+hh:=complexNormalize gg
+--R
+--R %pi
+--R (10) - ---
+--R 3
+--R 4a
+--R Type: Expression Complex
Integer
+--E
+
@
\section{\cite{1}:14.216~~~~~$\displaystyle\int{\sqrt{x^2-a^2}}~dx$}
@@ -341,7 +566,7 @@ $$
<<*>>=
)clear all
---S 24
+--S 39
aa:=integrate(sqrt(x^2-a^2),x)
--R
--R
@@ -360,7 +585,7 @@ aa:=integrate(sqrt(x^2-a^2),x)
--R Type: Union(Expression
Integer,...)
--E
---S 25
+--S 40
bb:=(x*sqrt(x^2-a^2))/2-a^2/2*log(x+sqrt(x^2-a^2))
--R
--R +-------+ +-------+
@@ -371,7 +596,7 @@ bb:=(x*sqrt(x^2-a^2))/2-a^2/2*log(x+sqrt(x^2-a^2))
--R Type: Expression
Integer
--E
---S 26 14:216 Axiom cannot simplify this expression
+--S 41
cc:=aa-bb
--R
--R +-------+ +-------+
@@ -382,6 +607,15 @@ cc:=aa-bb
--R Type: Expression
Integer
--E
+--S 42 14:216 Schaums and Axiom differ by a constant
+dd:=complexNormalize cc
+--R
+--R 2 2
+--R a log(- a )
+--R (4) -----------
+--R 2
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.217~~~~~$\displaystyle\int{x\sqrt{x^2-a^2}}~dx$}
@@ -391,7 +625,7 @@ $$
<<*>>=
)clear all
---S 27
+--S 43
aa:=integrate(x*sqrt(x^2-a^2),x)
--R
--R
@@ -405,7 +639,7 @@ aa:=integrate(x*sqrt(x^2-a^2),x)
--R Type: Union(Expression
Integer,...)
--E
---S 28
+--S 44
bb:=(x^2-a^2)^(3/2)/3
--R
--R +-------+
@@ -416,7 +650,7 @@ bb:=(x^2-a^2)^(3/2)/3
--R Type: Expression
Integer
--E
---S 29 14:217 Schaums and Axiom agree
+--S 45 14:217 Schaums and Axiom agree
cc:=aa-bb
--R
--R (3) 0
@@ -433,7 +667,7 @@ $$
<<*>>=
)clear all
---S 30
+--S 46
aa:=integrate(x^2*sqrt(x^2-a^2),x)
--R
--R
@@ -452,7 +686,7 @@ aa:=integrate(x^2*sqrt(x^2-a^2),x)
--R Type: Union(Expression
Integer,...)
--E
---S 31
+--S 47
bb:=(x*(x^2-a^2)^(3/2))/4+(a^2*x*sqrt(x^2-a^2))/8-a^4/8*log(x+sqrt(x^2-a^2))
--R
--R +-------+ +-------+
@@ -463,7 +697,7 @@
bb:=(x*(x^2-a^2)^(3/2))/4+(a^2*x*sqrt(x^2-a^2))/8-a^4/8*log(x+sqrt(x^2-a^2))
--R Type: Expression
Integer
--E
---S 32 14:218 Axiom cannot simplify this expression
+--S 48
cc:=aa-bb
--R
--R +-------+ +-------+
@@ -473,6 +707,16 @@ cc:=aa-bb
--R 8
--R Type: Expression
Integer
--E
+
+--S 49 14:218 Schaums and Axiom differ by a constant
+dd:=complexNormalize cc
+--R
+--R 4 2
+--R a log(- a )
+--R (4) -----------
+--R 8
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.219~~~~~$\displaystyle
@@ -483,7 +727,7 @@ $$
<<*>>=
)clear all
---S 33
+--S 50
aa:=integrate(x^3*sqrt(x^2-a^2),x)
--R
--R
@@ -501,7 +745,7 @@ aa:=integrate(x^3*sqrt(x^2-a^2),x)
--R Type: Union(Expression
Integer,...)
--E
---S 34
+--S 51
bb:=(x^2-a^2)^(5/2)/5+(a^2*(x^2-a^2)^(3/2))/3
--R
--R +-------+
@@ -512,7 +756,7 @@ bb:=(x^2-a^2)^(5/2)/5+(a^2*(x^2-a^2)^(3/2))/3
--R Type: Expression
Integer
--E
---S 35 14:219 Schaums and Axiom agree
+--S 52 14:219 Schaums and Axiom agree
cc:=aa-bb
--R
--R (3) 0
@@ -528,7 +772,7 @@ $$
<<*>>=
)clear all
---S 36
+--S 53
aa:=integrate(sqrt(x^2-a^2)/x,x)
--R
--R
@@ -544,26 +788,129 @@ aa:=integrate(sqrt(x^2-a^2)/x,x)
--R Type: Union(Expression
Integer,...)
--E
---S 37
-bb:=sqrt(x^2-a^2)-a*asec(abs(x/a))
+--S 54
+bb:=sqrt(x^2-a^2)-a*asec(x/a)
--R
---R +-------+
---R x | 2 2
---R (2) - a asec(abs(-)) + \|x - a
---R a
+--R +-------+
+--R | 2 2 x
+--R (2) \|x - a - a asec(-)
+--R a
--R Type: Expression
Integer
--E
---S 38 14:220 Axiom cannot simplify this expression
+--S 55
cc:=aa-bb
--R
--R +-------+
--R | 2 2
---R \|x - a - x x
---R (3) - 2a atan(--------------) + a asec(abs(-))
---R a a
+--R \|x - a - x x
+--R (3) - 2a atan(--------------) + a asec(-)
+--R a a
--R Type: Expression
Integer
--E
+
+--S 56
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R - x + %i
+--R %i log(--------)
+--R x + %i
+--R (4) atan(x) == - ----------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 57
+dd:=atanrule cc
+--R
+--R +-------+
+--R | 2 2
+--R - \|x - a + x + %i a x
+--R (5) %i a log(-----------------------) + a asec(-)
+--R +-------+ a
+--R | 2 2
+--R \|x - a - x + %i a
+--R Type: Expression Complex
Integer
+--E
+
+--S 58
+asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x))
+--R
+--R +------+
+--R | 2
+--R |x - 1
+--R x |------ + %i
+--R | 2
+--R \| x
+--R 2%i log(---------------) + %pi
+--R x
+--R (6) asec(x) == ------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 59
+ee:=asecrule dd
+--R
+--R (7)
+--R +-------+
+--R | 2 2
+--R |x - a
+--R x |------- + %i a +-------+
+--R | 2 | 2 2
+--R \| x - \|x - a + x + %i a
+--R 2%i a log(------------------) + 2%i a log(-----------------------) + a
%pi
+--R x +-------+
+--R | 2 2
+--R \|x - a - x + %i a
+--R
--------------------------------------------------------------------------
+--R 2
+--R Type: Expression Complex
Integer
+--E
+
+--S 60
+ff:=expandLog ee
+--R
+--R (8)
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R - 2%i a log(\|x - a - x + %i a) + 2%i a log(\|x - a - x - %i a)
+--R +
+--R +-------+
+--R | 2 2
+--R |x - a
+--R 2%i a log(x |------- + %i a) - 2%i a log(x) + 2%i a log(- 1) + a %pi
+--R | 2
+--R \| x
+--R /
+--R 2
+--R Type: Expression Complex
Integer
+--E
+
+--S 61
+gg:=rootSimp ff
+--R
+--R (9)
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R 2%i a log(\|x - a + %i a) - 2%i a log(\|x - a - x + %i a)
+--R +
+--R +-------+
+--R | 2 2
+--R 2%i a log(\|x - a - x - %i a) - 2%i a log(x) + 2%i a log(- 1) + a
%pi
+--R /
+--R 2
+--R Type: Expression Complex
Integer
+--E
+
+--S 62 14:220 Schaums and Axiom differ by a constant
+hh:=complexNormalize gg
+--R
+--R a %pi
+--R (10) -----
+--R 2
+--R Type: Expression Complex
Integer
+--E
@
\section{\cite{1}:14.221~~~~~$\displaystyle
@@ -574,7 +921,7 @@ $$
<<*>>=
)clear all
---S 39
+--S 63
aa:=integrate(sqrt(x^2-a^2)/x^2,x)
--R
--R
@@ -588,7 +935,7 @@ aa:=integrate(sqrt(x^2-a^2)/x^2,x)
--R Type: Union(Expression
Integer,...)
--E
---S 40
+--S 64
bb:=-sqrt(x^2-a^2)/x+log(x+sqrt(x^2-a^2))
--R
--R +-------+ +-------+
@@ -599,7 +946,7 @@ bb:=-sqrt(x^2-a^2)/x+log(x+sqrt(x^2-a^2))
--R Type: Expression
Integer
--E
---S 41 14:221 Axiom cannot simplify this expression
+--S 65
cc:=aa-bb
--R
--R +-------+ +-------+
@@ -608,6 +955,14 @@ cc:=aa-bb
--R Type: Expression
Integer
--E
+--S 66 14:221 Schaums and Axiom differ by a constant
+dd:=complexNormalize cc
+--R
+--R 2
+--R (4) - log(- a ) - 1
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.222~~~~~$\displaystyle
@@ -619,7 +974,7 @@ $$
<<*>>=
)clear all
---S 42
+--S 67
aa:=integrate(sqrt(x^2-a^2)/x^3,x)
--R
--R
@@ -640,31 +995,136 @@ aa:=integrate(sqrt(x^2-a^2)/x^3,x)
--R Type: Union(Expression
Integer,...)
--E
---S 43
-bb:=-sqrt(x^2-a^2)/(2*x^2)+1/(2*a)*asec(abs(x/a))
+--S 68
+bb:=-sqrt(x^2-a^2)/(2*x^2)+1/(2*a)*asec(x/a)
--R
---R +-------+
---R 2 x | 2 2
---R x asec(abs(-)) - a\|x - a
---R a
---R (2) ----------------------------
---R 2
---R 2a x
+--R +-------+
+--R | 2 2 2 x
+--R - a\|x - a + x asec(-)
+--R a
+--R (2) -------------------------
+--R 2
+--R 2a x
--R Type: Expression
Integer
--E
---S 44 14:222 Axiom cannot simplify this expression
+--S 69
cc:=aa-bb
--R
--R +-------+
--R | 2 2
---R \|x - a - x x
---R 2atan(--------------) - asec(abs(-))
---R a a
---R (3) ------------------------------------
---R 2a
+--R \|x - a - x x
+--R 2atan(--------------) - asec(-)
+--R a a
+--R (3) -------------------------------
+--R 2a
--R Type: Expression
Integer
--E
+
+--S 70
+asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x))
+--R
+--R +------+
+--R | 2
+--R |x - 1
+--R x |------ + %i
+--R | 2
+--R \| x
+--R 2%i log(---------------) + %pi
+--R x
+--R (4) asec(x) == ------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 71
+dd:=asecrule cc
+--R
+--R +-------+
+--R | 2 2
+--R |x - a
+--R x |------- + %i a +-------+
+--R | 2 | 2 2
+--R \| x \|x - a - x
+--R - 2%i log(------------------) + 4atan(--------------) - %pi
+--R x a
+--R (5) -----------------------------------------------------------
+--R 4a
+--R Type: Expression Complex
Integer
+--E
+
+--S 72
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R - x + %i
+--R %i log(--------)
+--R x + %i
+--R (6) atan(x) == - ----------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 73
+ee:=atanrule dd
+--R
+--R +-------+
+--R | 2 2
+--R |x - a
+--R x |------- + %i a +-------+
+--R | 2 | 2 2
+--R \| x - \|x - a + x + %i a
+--R - 2%i log(------------------) - 2%i log(-----------------------) -
%pi
+--R x +-------+
+--R | 2 2
+--R \|x - a - x + %i a
+--R (7)
----------------------------------------------------------------------
+--R 4a
+--R Type: Expression Complex
Integer
+--E
+
+--S 74
+ff:=expandLog ee
+--R
+--R (8)
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R 2%i log(\|x - a - x + %i a) - 2%i log(\|x - a - x - %i a)
+--R +
+--R +-------+
+--R | 2 2
+--R |x - a
+--R - 2%i log(x |------- + %i a) + 2%i log(x) - 2%i log(- 1) - %pi
+--R | 2
+--R \| x
+--R /
+--R 4a
+--R Type: Expression Complex
Integer
+--E
+
+--S 75
+gg:=rootSimp ff
+--R
+--R (9)
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R - 2%i log(\|x - a + %i a) + 2%i log(\|x - a - x + %i a)
+--R +
+--R +-------+
+--R | 2 2
+--R - 2%i log(\|x - a - x - %i a) + 2%i log(x) - 2%i log(- 1) - %pi
+--R /
+--R 4a
+--R Type: Expression Complex
Integer
+--E
+
+--S 76 14:222 Schaums and Axiom differ by a constant
+hh:=complexNormalize gg
+--R
+--R %pi
+--R (10) - ---
+--R 4a
+--R Type: Expression Complex
Integer
+--E
@
\section{\cite{1}:14.223~~~~~$\displaystyle\int{\frac{dx}{(x^2-a^2)^{3/2}}}$}
@@ -674,7 +1134,7 @@ $$
<<*>>=
)clear all
---S 45
+--S 77
aa:=integrate(1/(x^2-a^2)^(3/2),x)
--R
--R
@@ -686,7 +1146,7 @@ aa:=integrate(1/(x^2-a^2)^(3/2),x)
--R Type: Union(Expression
Integer,...)
--E
---S 46
+--S 78
bb:=-x/(a^2*sqrt(x^2-a^2))
--R
--R x
@@ -697,7 +1157,7 @@ bb:=-x/(a^2*sqrt(x^2-a^2))
--R Type: Expression
Integer
--E
---S 47 14:223 Schaums and Axiom differ by a constant
+--S 79 14:223 Schaums and Axiom differ by a constant
cc:=aa-bb
--R
--R 1
@@ -716,7 +1176,7 @@ $$
<<*>>=
)clear all
---S 48
+--S 80
aa:=integrate(x/(x^2-a^2)^(3/2),x)
--R
--R
@@ -730,7 +1190,7 @@ aa:=integrate(x/(x^2-a^2)^(3/2),x)
--R Type: Union(Expression
Integer,...)
--E
---S 49
+--S 81
bb:=-1/sqrt(x^2-a^2)
--R
--R 1
@@ -741,7 +1201,7 @@ bb:=-1/sqrt(x^2-a^2)
--R Type: Expression
Integer
--E
---S 50 14:224 Schaums and Axiom agree
+--S 82 14:224 Schaums and Axiom agree
cc:=aa-bb
--R
--R (3) 0
@@ -757,7 +1217,7 @@ $$
<<*>>=
)clear all
---S 51
+--S 83
aa:=integrate(x^2/(x^2-a^2)^(3/2),x)
--R
--R
@@ -771,7 +1231,7 @@ aa:=integrate(x^2/(x^2-a^2)^(3/2),x)
--R Type: Union(Expression
Integer,...)
--E
---S 52
+--S 84
bb:=-x/sqrt(x^2-a^2)+log(x+sqrt(x^2-a^2))
--R
--R +-------+ +-------+
@@ -784,7 +1244,7 @@ bb:=-x/sqrt(x^2-a^2)+log(x+sqrt(x^2-a^2))
--R Type: Expression
Integer
--E
---S 53 14:225 Axiom cannot simplify this expression
+--S 85
cc:=aa-bb
--R
--R +-------+ +-------+
@@ -792,6 +1252,14 @@ cc:=aa-bb
--R (3) - log(\|x - a + x) - log(\|x - a - x) - 1
--R Type: Expression
Integer
--E
+
+--S 86 14:225 Schaums and Axiom differ by a constant
+dd:=complexNormalize cc
+--R
+--R 2
+--R (4) - log(- a ) - 1
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.226~~~~~$\displaystyle
@@ -802,7 +1270,7 @@ $$
<<*>>=
)clear all
---S 54
+--S 87
aa:=integrate(x^3/(x^2-a^2)^(3/2),x)
--R
--R
@@ -816,7 +1284,7 @@ aa:=integrate(x^3/(x^2-a^2)^(3/2),x)
--R Type: Union(Expression
Integer,...)
--E
---S 55
+--S 88
bb:=sqrt(x^2-a^2)-a^2/sqrt(x^2-a^2)
--R
--R 2 2
@@ -828,7 +1296,7 @@ bb:=sqrt(x^2-a^2)-a^2/sqrt(x^2-a^2)
--R Type: Expression
Integer
--E
---S 56 14:226 Schaums and Axiom agree
+--S 89 14:226 Schaums and Axiom agree
cc:=aa-bb
--R
--R (3) 0
@@ -845,7 +1313,7 @@ $$
<<*>>=
)clear all
---S 57
+--S 90
aa:=integrate(1/(x*(x^2-a^2)^(3/2)),x)
--R
--R
@@ -861,33 +1329,143 @@ aa:=integrate(1/(x*(x^2-a^2)^(3/2)),x)
--R Type: Union(Expression
Integer,...)
--E
---S 58
-bb:=-1/(a^2*sqrt(x^2-a^2))-1/a^3*asec(abs(x/a))
+--S 91
+bb:=-1/(a^2*sqrt(x^2-a^2))-1/a^3*asec(x/a)
--R
---R +-------+
---R | 2 2 x
---R - \|x - a asec(abs(-)) - a
---R a
---R (2) ----------------------------
---R +-------+
---R 3 | 2 2
---R a \|x - a
+--R +-------+
+--R x | 2 2
+--R - asec(-)\|x - a - a
+--R a
+--R (2) -----------------------
+--R +-------+
+--R 3 | 2 2
+--R a \|x - a
--R Type: Expression
Integer
--E
---S 59 14:227 Axiom cannot simplify this expression
+--S 92
cc:=aa-bb
--R
--R +-------+
--R | 2 2
---R \|x - a - x x
---R - 2atan(--------------) + asec(abs(-))
---R a a
---R (3) --------------------------------------
---R 3
---R a
+--R \|x - a - x x
+--R - 2atan(--------------) + asec(-)
+--R a a
+--R (3) ---------------------------------
+--R 3
+--R a
--R Type: Expression
Integer
--E
+
+--S 93
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R - x + %i
+--R %i log(--------)
+--R x + %i
+--R (4) atan(x) == - ----------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 94
+dd:=atanrule cc
+--R
+--R +-------+
+--R | 2 2
+--R - \|x - a + x + %i a x
+--R %i log(-----------------------) + asec(-)
+--R +-------+ a
+--R | 2 2
+--R \|x - a - x + %i a
+--R (5) -----------------------------------------
+--R 3
+--R a
+--R Type: Expression Complex
Integer
+--E
+
+--S 95
+asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x))
+--R
+--R +------+
+--R | 2
+--R |x - 1
+--R x |------ + %i
+--R | 2
+--R \| x
+--R 2%i log(---------------) + %pi
+--R x
+--R (6) asec(x) == ------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 96
+ee:=asecrule dd
+--R
+--R +-------+
+--R | 2 2
+--R |x - a
+--R x |------- + %i a +-------+
+--R | 2 | 2 2
+--R \| x - \|x - a + x + %i a
+--R 2%i log(------------------) + 2%i log(-----------------------) + %pi
+--R x +-------+
+--R | 2 2
+--R \|x - a - x + %i a
+--R (7) --------------------------------------------------------------------
+--R 3
+--R 2a
+--R Type: Expression Complex
Integer
+--E
+
+--S 97
+ff:=expandLog ee
+--R
+--R (8)
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R - 2%i log(\|x - a - x + %i a) + 2%i log(\|x - a - x - %i a)
+--R +
+--R +-------+
+--R | 2 2
+--R |x - a
+--R 2%i log(x |------- + %i a) - 2%i log(x) + 2%i log(- 1) + %pi
+--R | 2
+--R \| x
+--R /
+--R 3
+--R 2a
+--R Type: Expression Complex
Integer
+--E
+
+--S 98
+gg:=rootSimp ff
+--R
+--R (9)
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R 2%i log(\|x - a + %i a) - 2%i log(\|x - a - x + %i a)
+--R +
+--R +-------+
+--R | 2 2
+--R 2%i log(\|x - a - x - %i a) - 2%i log(x) + 2%i log(- 1) + %pi
+--R /
+--R 3
+--R 2a
+--R Type: Expression Complex
Integer
+--E
+
+--S 99 14:227 Schaums and Axiom differ by a constant
+hh:=complexNormalize gg
+--R
+--R %pi
+--R (10) ---
+--R 3
+--R 2a
+--R Type: Expression Complex
Integer
+--E
+
@
\section{\cite{1}:14.228~~~~~$\displaystyle
@@ -898,7 +1476,7 @@ $$
<<*>>=
)clear all
---S 60
+--S 100
aa:=integrate(1/(x^2*(x^2-a^2)^(3/2)),x)
--R
--R
@@ -910,7 +1488,7 @@ aa:=integrate(1/(x^2*(x^2-a^2)^(3/2)),x)
--R Type: Union(Expression
Integer,...)
--E
---S 61
+--S 101
bb:=-sqrt(x^2-a^2)/(a^4*x)-x/(a^4*sqrt(x^2-a^2))
--R
--R 2 2
@@ -922,7 +1500,7 @@ bb:=-sqrt(x^2-a^2)/(a^4*x)-x/(a^4*sqrt(x^2-a^2))
--R Type: Expression
Integer
--E
---S 62 14:228 Schaums and Axiom differ by a constant
+--S 102 14:228 Schaums and Axiom differ by a constant
cc:=aa-bb
--R
--R 2
@@ -944,7 +1522,7 @@ $$
<<*>>=
)clear all
---S 63
+--S 103
aa:=integrate(1/(x^3*(x^2-a^2)^(3/2)),x)
--R
--R
@@ -965,35 +1543,143 @@ aa:=integrate(1/(x^3*(x^2-a^2)^(3/2)),x)
--R Type: Union(Expression
Integer,...)
--E
---S 64
-bb:=1/(2*a^2*x^2*sqrt(x^2-a^2))-3/(2*a^4*sqrt(x^2-a^2))-3/(2*a^5)*asec(abs(x/a))
+--S 104
+bb:=1/(2*a^2*x^2*sqrt(x^2-a^2))-3/(2*a^4*sqrt(x^2-a^2))-3/(2*a^5)*asec(x/a)
--R
---R +-------+
---R 2 | 2 2 x 2 3
---R - 3x \|x - a asec(abs(-)) - 3a x + a
---R a
---R (2) ----------------------------------------
---R +-------+
---R 5 2 | 2 2
---R 2a x \|x - a
+--R +-------+
+--R 2 x | 2 2 2 3
+--R - 3x asec(-)\|x - a - 3a x + a
+--R a
+--R (2) -----------------------------------
+--R +-------+
+--R 5 2 | 2 2
+--R 2a x \|x - a
--R Type: Expression
Integer
--E
---S 65 14:229 Axiom cannot simplify this expression
+--S 105
cc:=aa-bb
--R
--R +-------+
--R | 2 2
---R \|x - a - x x
---R - 6atan(--------------) + 3asec(abs(-))
---R a a
---R (3) ---------------------------------------
---R 5
---R 2a
+--R \|x - a - x x
+--R - 6atan(--------------) + 3asec(-)
+--R a a
+--R (3) ----------------------------------
+--R 5
+--R 2a
--R Type: Expression
Integer
--E
-@
+--S 106
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R - x + %i
+--R %i log(--------)
+--R x + %i
+--R (4) atan(x) == - ----------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 107
+dd:=atanrule cc
+--R
+--R +-------+
+--R | 2 2
+--R - \|x - a + x + %i a x
+--R 3%i log(-----------------------) + 3asec(-)
+--R +-------+ a
+--R | 2 2
+--R \|x - a - x + %i a
+--R (5) -------------------------------------------
+--R 5
+--R 2a
+--R Type: Expression Complex
Integer
+--E
+
+--S 108
+asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x))
+--R
+--R +------+
+--R | 2
+--R |x - 1
+--R x |------ + %i
+--R | 2
+--R \| x
+--R 2%i log(---------------) + %pi
+--R x
+--R (6) asec(x) == ------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 109
+ee:=asecrule dd
+--R
+--R +-------+
+--R | 2 2
+--R |x - a
+--R x |------- + %i a +-------+
+--R | 2 | 2 2
+--R \| x - \|x - a + x + %i a
+--R 6%i log(------------------) + 6%i log(-----------------------) +
3%pi
+--R x +-------+
+--R | 2 2
+--R \|x - a - x + %i a
+--R (7)
---------------------------------------------------------------------
+--R 5
+--R 4a
+--R Type: Expression Complex
Integer
+--E
+
+--S 110
+ff:=expandLog ee
+--R
+--R (8)
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R - 6%i log(\|x - a - x + %i a) + 6%i log(\|x - a - x - %i a)
+--R +
+--R +-------+
+--R | 2 2
+--R |x - a
+--R 6%i log(x |------- + %i a) - 6%i log(x) + 6%i log(- 1) + 3%pi
+--R | 2
+--R \| x
+--R /
+--R 5
+--R 4a
+--R Type: Expression Complex
Integer
+--E
+
+--S 111
+gg:=rootSimp ff
+--R
+--R (9)
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R 6%i log(\|x - a + %i a) - 6%i log(\|x - a - x + %i a)
+--R +
+--R +-------+
+--R | 2 2
+--R 6%i log(\|x - a - x - %i a) - 6%i log(x) + 6%i log(- 1) + 3%pi
+--R /
+--R 5
+--R 4a
+--R Type: Expression Complex
Integer
+--E
+
+--S 112 14:229 Schaums and Axiom differ by a constant
+hh:=complexNormalize gg
+--R
+--R 3%pi
+--R (10) ----
+--R 5
+--R 4a
+--R Type: Expression Complex
Integer
+--E
+@
\section{\cite{1}:14.230~~~~~$\displaystyle\int{(x^2-a^2)^{3/2}}~dx$}
$$\int{(x^2-a^2)^{3/2}}=
\frac{x(x^2-a^2)^{3/2}}{4}-\frac{3a^2x\sqrt{x^2-a^2}}{8}+
@@ -1002,7 +1688,7 @@ $$
<<*>>=
)clear all
---S 66
+--S 113
aa:=integrate((x^2-a^2)^(3/2),x)
--R
--R
@@ -1024,7 +1710,7 @@ aa:=integrate((x^2-a^2)^(3/2),x)
--R Type: Union(Expression
Integer,...)
--E
---S 67
+--S 114
bb:=(x*(x^2-a^2)^(3/2))/4-(3*a^2*x*sqrt(x^2-a^2))/8+3/8*a^4*log(x+sqrt(x^2-a^2))
--R
--R +-------+ +-------+
@@ -1035,7 +1721,7 @@
bb:=(x*(x^2-a^2)^(3/2))/4-(3*a^2*x*sqrt(x^2-a^2))/8+3/8*a^4*log(x+sqrt(x^2-a^2))
--R Type: Expression
Integer
--E
---S 68 14:230 Axiom cannot simplify this expression
+--S 115
cc:=aa-bb
--R
--R +-------+ +-------+
@@ -1046,6 +1732,16 @@ cc:=aa-bb
--R Type: Expression
Integer
--E
+--S 116 14:230 Schaums and Axiom differ by a constant
+dd:=complexNormalize cc
+--R
+--R 4 2
+--R 3a log(- a )
+--R (4) - ------------
+--R 8
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.231~~~~~$\displaystyle\int{x(x^2-a^2)^{3/2}}~dx$}
@@ -1053,7 +1749,7 @@ $$\int{x(x^2-a^2)^{3/2}}=\frac{(x^2-a^2)^{5/2}}{5}$$
<<*>>=
)clear all
---S 69
+--S 117
aa:=integrate(x*(x^2-a^2)^(3/2),x)
--R
--R
@@ -1071,7 +1767,7 @@ aa:=integrate(x*(x^2-a^2)^(3/2),x)
--R Type: Union(Expression
Integer,...)
--E
---S 70
+--S 118
bb:=(x^2-a^2)^(5/2)/5
--R
--R +-------+
@@ -1082,7 +1778,7 @@ bb:=(x^2-a^2)^(5/2)/5
--R Type: Expression
Integer
--E
---S 71 14:231 Schaums and Axiom agree
+--S 119 14:231 Schaums and Axiom agree
cc:=aa-bb
--R
--R (3) 0
@@ -1099,7 +1795,7 @@ $$
<<*>>=
)clear all
---S 72
+--S 120
aa:=integrate(x^2*(x^2-a^2)^(3/2),x)
--R
--R
@@ -1128,7 +1824,7 @@ aa:=integrate(x^2*(x^2-a^2)^(3/2),x)
--R Type: Union(Expression
Integer,...)
--E
---S 73
+--S 121
bb:=(x*(x^2-a^2)^(5/2))/6+(a^2*x*(x^2-a^2)^(3/2))/24-(a^4*x*sqrt(x^2-a^2))/16+a^6/16*log(x+sqrt(x^2-a^2))
--R
--R +-------+ +-------+
@@ -1139,7 +1835,7 @@
bb:=(x*(x^2-a^2)^(5/2))/6+(a^2*x*(x^2-a^2)^(3/2))/24-(a^4*x*sqrt(x^2-a^2))/16+a^
--R Type: Expression
Integer
--E
---S 74 14:232 Axiom cannot simplify this expression
+--S 122
cc:=aa-bb
--R
--R +-------+ +-------+
@@ -1149,6 +1845,16 @@ cc:=aa-bb
--R 16
--R Type: Expression
Integer
--E
+
+--S 123 14:232 Schaums and Axiom differ by a constant
+dd:=complexNormalize cc
+--R
+--R 6 2
+--R a log(- a )
+--R (4) - -----------
+--R 16
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.233~~~~~$\displaystyle\int{x^3(x^2-a^2)^{3/2}}~dx$}
@@ -1158,7 +1864,7 @@ $$
<<*>>=
)clear all
---S 75
+--S 124
aa:=integrate(x^3*(x^2-a^2)^(3/2),x)
--R
--R
@@ -1188,7 +1894,7 @@ aa:=integrate(x^3*(x^2-a^2)^(3/2),x)
--R Type: Union(Expression
Integer,...)
--E
---S 76
+--S 125
bb:=(x^2-a^2)^(7/2)/7+(a^2*(x^2-a^2)^(5/2))/5
--R
--R +-------+
@@ -1199,7 +1905,7 @@ bb:=(x^2-a^2)^(7/2)/7+(a^2*(x^2-a^2)^(5/2))/5
--R Type: Expression
Integer
--E
---S 77 14:233 Schaums and Axiom agree
+--S 126 14:233 Schaums and Axiom agree
cc:=aa-bb
--R
--R (3) 0
@@ -1216,7 +1922,7 @@ $$
<<*>>=
)clear all
---S 78
+--S 127
aa:=integrate((x^2-a^2)^(3/2)/x,x)
--R
--R
@@ -1237,28 +1943,135 @@ aa:=integrate((x^2-a^2)^(3/2)/x,x)
--R Type: Union(Expression
Integer,...)
--E
---S 79
-bb:=(x^2-a^2)^(3/2)/3-a^2*sqrt(x^2-a^2)+a^3*asec(abs(x/a))
+--S 128
+bb:=(x^2-a^2)^(3/2)/3-a^2*sqrt(x^2-a^2)+a^3*asec(x/a)
--R
---R +-------+
---R 3 x 2 2 | 2 2
---R 3a asec(abs(-)) + (x - 4a )\|x - a
---R a
---R (2) --------------------------------------
---R 3
+--R +-------+
+--R 2 2 | 2 2 3 x
+--R (x - 4a )\|x - a + 3a asec(-)
+--R a
+--R (2) ---------------------------------
+--R 3
--R Type: Expression
Integer
--E
---S 80 14:234 Axiom cannot simplify this expression
+--S 129
cc:=aa-bb
--R
--R +-------+
--R | 2 2
---R 3 \|x - a - x 3 x
---R (3) 2a atan(--------------) - a asec(abs(-))
---R a a
+--R 3 \|x - a - x 3 x
+--R (3) 2a atan(--------------) - a asec(-)
+--R a a
--R Type: Expression
Integer
--E
+
+--S 130
+asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x))
+--R
+--R +------+
+--R | 2
+--R |x - 1
+--R x |------ + %i
+--R | 2
+--R \| x
+--R 2%i log(---------------) + %pi
+--R x
+--R (4) asec(x) == ------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 131
+dd:=asecrule cc
+--R
+--R +-------+
+--R | 2 2
+--R |x - a
+--R x |------- + %i a +-------+
+--R | 2 | 2 2
+--R 3 \| x 3 \|x - a - x 3
+--R - 2%i a log(------------------) + 4a atan(--------------) - a %pi
+--R x a
+--R (5) -----------------------------------------------------------------
+--R 2
+--R Type: Expression Complex
Integer
+--E
+
+--S 132
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R - x + %i
+--R %i log(--------)
+--R x + %i
+--R (6) atan(x) == - ----------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 133
+ee:=atanrule dd
+--R
+--R (7)
+--R +-------+
+--R | 2 2
+--R |x - a
+--R x |------- + %i a +-------+
+--R | 2 | 2 2
+--R 3 \| x 3 - \|x - a + x + %i a 3
+--R - 2%i a log(------------------) - 2%i a log(-----------------------) - a
%pi
+--R x +-------+
+--R | 2 2
+--R \|x - a - x + %i a
+--R
----------------------------------------------------------------------------
+--R 2
+--R Type: Expression Complex
Integer
+--E
+
+--S 134
+ff:=expandLog ee
+--R
+--R (8)
+--R +-------+ +-------+
+--R 3 | 2 2 3 | 2 2
+--R 2%i a log(\|x - a - x + %i a) - 2%i a log(\|x - a - x - %i a)
+--R +
+--R +-------+
+--R | 2 2
+--R 3 |x - a 3 3 3
+--R - 2%i a log(x |------- + %i a) + 2%i a log(x) - 2%i a log(- 1) - a
%pi
+--R | 2
+--R \| x
+--R /
+--R 2
+--R Type: Expression Complex
Integer
+--E
+
+--S 135
+gg:=rootSimp ff
+--R
+--R (9)
+--R +-------+ +-------+
+--R 3 | 2 2 3 | 2 2
+--R - 2%i a log(\|x - a + %i a) + 2%i a log(\|x - a - x + %i a)
+--R +
+--R +-------+
+--R 3 | 2 2 3 3 3
+--R - 2%i a log(\|x - a - x - %i a) + 2%i a log(x) - 2%i a log(- 1) - a
%pi
+--R /
+--R 2
+--R Type: Expression Complex
Integer
+--E
+
+--S 136 14:234 Schaums and Axiom differ by a constant
+hh:=complexNormalize gg
+--R
+--R 3
+--R a %pi
+--R (10) - -----
+--R 2
+--R Type: Expression Complex
Integer
+--E
@
\section{\cite{1}:14.235~~~~~$\displaystyle
@@ -1270,7 +2083,7 @@ $$
<<*>>=
)clear all
---S 81
+--S 137
aa:=integrate((x^2-a^2)^{3/2}/x^2,x)
--R
--R
@@ -1289,7 +2102,7 @@ aa:=integrate((x^2-a^2)^{3/2}/x^2,x)
--R Type: Union(Expression
Integer,...)
--E
---S 82
+--S 138
bb:=-(x^2-a^2)^(3/2)/x+3*x*sqrt(x^2-a^2)/2-3/2*a^2*log(x+sqrt(x^2-a^2))
--R
--R +-------+ +-------+
@@ -1300,7 +2113,7 @@
bb:=-(x^2-a^2)^(3/2)/x+3*x*sqrt(x^2-a^2)/2-3/2*a^2*log(x+sqrt(x^2-a^2))
--R Type: Expression
Integer
--E
---S 83 14:235 Axiom cannot simplify this expression
+--S 139
cc:=aa-bb
--R
--R +-------+ +-------+
@@ -1310,6 +2123,17 @@ cc:=aa-bb
--R 2
--R Type: Expression
Integer
--E
+
+--S 140 14:235 Schaums and Axiom differ by a constant
+dd:=complexNormalize cc
+--R
+--R 2 2 2
+--R 3a log(- a ) + 2a
+--R (4) ------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.236~~~~~$\displaystyle
@@ -1321,7 +2145,7 @@ $$
<<*>>=
)clear all
---S 84
+--S 141
aa:=integrate((x^2-a^2)^(3/2)/x^3,x)
--R
--R
@@ -1342,31 +2166,137 @@ aa:=integrate((x^2-a^2)^(3/2)/x^3,x)
--R Type: Union(Expression
Integer,...)
--E
---S 85
-bb:=-(x^2-a^2)^(3/2)/(2*x^2)+(3*sqrt(x^2-a^2))/2-3/2*a*asec(abs(x/a))
+--S 142
+bb:=-(x^2-a^2)^(3/2)/(2*x^2)+(3*sqrt(x^2-a^2))/2-3/2*a*asec(x/a)
--R
---R +-------+
---R 2 x 2 2 | 2 2
---R - 3a x asec(abs(-)) + (2x + a )\|x - a
---R a
---R (2) ------------------------------------------
---R 2
---R 2x
+--R +-------+
+--R 2 2 | 2 2 2 x
+--R (2x + a )\|x - a - 3a x asec(-)
+--R a
+--R (2) -----------------------------------
+--R 2
+--R 2x
--R Type: Expression
Integer
--E
---S 86 14:236 Axiom cannot simplify this expression
+--S 143
cc:=aa-bb
--R
--R +-------+
--R | 2 2
---R \|x - a - x x
---R - 6a atan(--------------) + 3a asec(abs(-))
---R a a
---R (3) -------------------------------------------
---R 2
+--R \|x - a - x x
+--R - 6a atan(--------------) + 3a asec(-)
+--R a a
+--R (3) --------------------------------------
+--R 2
--R Type: Expression
Integer
--E
+
+--S 144
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R - x + %i
+--R %i log(--------)
+--R x + %i
+--R (4) atan(x) == - ----------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 145
+dd:=atanrule cc
+--R
+--R +-------+
+--R | 2 2
+--R - \|x - a + x + %i a x
+--R 3%i a log(-----------------------) + 3a asec(-)
+--R +-------+ a
+--R | 2 2
+--R \|x - a - x + %i a
+--R (5) -----------------------------------------------
+--R 2
+--R Type: Expression Complex
Integer
+--E
+
+--S 146
+asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x))
+--R
+--R +------+
+--R | 2
+--R |x - 1
+--R x |------ + %i
+--R | 2
+--R \| x
+--R 2%i log(---------------) + %pi
+--R x
+--R (6) asec(x) == ------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 147
+ee:=asecrule dd
+--R
+--R (7)
+--R +-------+
+--R | 2 2
+--R |x - a
+--R x |------- + %i a +-------+
+--R | 2 | 2 2
+--R \| x - \|x - a + x + %i a
+--R 6%i a log(------------------) + 6%i a log(-----------------------) + 3a
%pi
+--R x +-------+
+--R | 2 2
+--R \|x - a - x + %i a
+--R
---------------------------------------------------------------------------
+--R 4
+--R Type: Expression Complex
Integer
+--E
+
+--S 148
+ff:=expandLog ee
+--R
+--R (8)
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R - 6%i a log(\|x - a - x + %i a) + 6%i a log(\|x - a - x - %i a)
+--R +
+--R +-------+
+--R | 2 2
+--R |x - a
+--R 6%i a log(x |------- + %i a) - 6%i a log(x) + 6%i a log(- 1) + 3a
%pi
+--R | 2
+--R \| x
+--R /
+--R 4
+--R Type: Expression Complex
Integer
+--E
+
+--S 149
+gg:=rootSimp ff
+--R
+--R (9)
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R 6%i a log(\|x - a + %i a) - 6%i a log(\|x - a - x + %i a)
+--R +
+--R +-------+
+--R | 2 2
+--R 6%i a log(\|x - a - x - %i a) - 6%i a log(x) + 6%i a log(- 1) + 3a
%pi
+--R /
+--R 4
+--R Type: Expression Complex
Integer
+--E
+
+--S 150 14:236 Schaums and Axiom differ by a constant
+hh:=complexNormalize gg
+--R
+--R 3a %pi
+--R (10) ------
+--R 4
+--R Type: Expression Complex
Integer
+--E
+
)spool
)lisp (bye)
@
diff --git a/src/input/schaum11.input.pamphlet
b/src/input/schaum11.input.pamphlet
index 6cd59a6..fc3117e 100644
--- a/src/input/schaum11.input.pamphlet
+++ b/src/input/schaum11.input.pamphlet
@@ -15,7 +15,7 @@
$$\int{\frac{1}{\sqrt{a^2-x^2}}}=\ln\left(x+\sqrt{a^2-x^2}\right)$$
)set message auto off
)clear all
---S 1 of 28
+--S 1
aa:=integrate(1/(sqrt(a^2-x^2)),x)
--R
--R
@@ -26,6 +26,96 @@ aa:=integrate(1/(sqrt(a^2-x^2)),x)
--R x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 2
+bb:=asin(x/a)
+--R
+--R x
+--R (2) asin(-)
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 3
+cc:=aa-bb
+--R
+--R +---------+
+--R | 2 2
+--R \|- x + a - a x
+--R (3) - 2atan(----------------) - asin(-)
+--R x a
+--R Type: Expression
Integer
+--E
+
+--S 4
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R - x + %i
+--R %i log(--------)
+--R x + %i
+--R (4) atan(x) == - ----------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 5
+dd:=atanrule cc
+--R
+--R +---------+
+--R | 2 2
+--R - \|- x + a + %i x + a x
+--R (5) %i log(-------------------------) - asin(-)
+--R +---------+ a
+--R | 2 2
+--R \|- x + a + %i x - a
+--R Type: Expression Complex
Integer
+--E
+
+--S 6
+asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2)))
+--R
+--R +--------+
+--R | 2
+--R (6) asin(x) == %i log(\|- x + 1 - %i x)
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 7
+ee:=asinrule dd
+--R
+--R +---------+
+--R | 2 2
+--R |- x + a
+--R a |--------- - %i x +---------+
+--R | 2 | 2 2
+--R \| a - \|- x + a + %i x + a
+--R (7) - %i log(--------------------) + %i log(-------------------------)
+--R a +---------+
+--R | 2 2
+--R \|- x + a + %i x - a
+--R Type: Expression Complex
Integer
+--E
+
+--S 8
+ff:=rootSimp ee
+--R
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R %i\|x - a - %i x - \|x - a + x - %i a
+--R (8) - %i log(-------------------) + %i log(-----------------------)
+--R a +-------+
+--R | 2 2
+--R \|x - a + x + %i a
+--R Type: Expression Complex
Integer
+--E
+
+--S 9 14:238 Schaums and Axiom agree
+gg:=complexNormalize ff
+--R
+--R (9) 0
+--R Type: Expression Complex
Integer
+--E
+
@
\section{\cite{1}:14.238~~~~~$\displaystyle\int{\frac{x~dx}{\sqrt{a^2-x^2}}}$}
@@ -33,7 +123,7 @@ $$\int{\frac{x}{\sqrt{a^2-x^2}}}=\sqrt{a^2-x^2}$$
<<*>>=
)clear all
---S 2 of 28
+--S 10
aa:=integrate(x/(sqrt(a^2-x^2)),x)
--R
--R
@@ -45,6 +135,22 @@ aa:=integrate(x/(sqrt(a^2-x^2)),x)
--R \|- x + a - a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 11
+bb:=-sqrt(a^2-x^2)
+--R
+--R +---------+
+--R | 2 2
+--R (2) - \|- x + a
+--R Type: Expression
Integer
+--E
+
+--S 12 14:238 Schaums and Axiom differ by a constant
+cc:=aa-bb
+--R
+--R (3) - a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.239~~~~~$\displaystyle
@@ -55,7 +161,7 @@ $$
<<*>>=
)clear all
---S 3 of 28
+--S 13
aa:=integrate(x^2/sqrt(a^2-x^2),x)
--R
--R
@@ -75,6 +181,127 @@ aa:=integrate(x^2/sqrt(a^2-x^2),x)
--R 4a\|- x + a + 2x - 4a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 14
+bb:=-(x*sqrt(a^2-x^2))/2+a^2/2*asin(x/a)
+--R
+--R +---------+
+--R | 2 2 2 x
+--R - x\|- x + a + a asin(-)
+--R a
+--R (2) ---------------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 15
+cc:=aa-bb
+--R
+--R +---------+
+--R | 2 2
+--R 2 \|- x + a - a 2 x
+--R - 2a atan(----------------) - a asin(-)
+--R x a
+--R (3) ---------------------------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 16
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R - x + %i
+--R %i log(--------)
+--R x + %i
+--R (4) atan(x) == - ----------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 17
+dd:=atanrule cc
+--R
+--R +---------+
+--R | 2 2
+--R 2 - \|- x + a + %i x + a 2 x
+--R %i a log(-------------------------) - a asin(-)
+--R +---------+ a
+--R | 2 2
+--R \|- x + a + %i x - a
+--R (5) -----------------------------------------------
+--R 2
+--R Type: Expression Complex
Integer
+--E
+
+--S 18
+asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2)))
+--R
+--R +--------+
+--R | 2
+--R (6) asin(x) == %i log(\|- x + 1 - %i x)
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 19
+ee:=asinrule dd
+--R
+--R +---------+
+--R | 2 2
+--R |- x + a
+--R a |--------- - %i x +---------+
+--R | 2 | 2 2
+--R 2 \| a 2 - \|- x + a + %i x +
a
+--R - %i a log(--------------------) + %i a
log(-------------------------)
+--R a +---------+
+--R | 2 2
+--R \|- x + a + %i x - a
+--R (7)
----------------------------------------------------------------------
+--R 2
+--R Type: Expression Complex
Integer
+--E
+
+--S 20
+ff:=expandLog ee
+--R
+--R (8)
+--R +---------+
+--R | 2 2 +---------+
+--R 2 |- x + a 2 | 2 2
+--R - %i a log(a |--------- - %i x) - %i a log(\|- x + a + %i x - a)
+--R | 2
+--R \| a
+--R +
+--R +---------+
+--R 2 | 2 2 2 2
+--R %i a log(\|- x + a - %i x - a) + %i a log(a) + %i a log(- 1)
+--R /
+--R 2
+--R Type: Expression Complex
Integer
+--E
+
+--S 21
+gg:=rootSimp ff
+--R
+--R (9)
+--R +-------+ +-------+
+--R 2 | 2 2 2 | 2 2
+--R - %i a log(%i\|x - a + %i x - a) - %i a log(%i\|x - a - %i x)
+--R +
+--R +-------+
+--R 2 | 2 2 2 2
+--R %i a log(%i\|x - a - %i x - a) + %i a log(a) + %i a log(- 1)
+--R /
+--R 2
+--R Type: Expression Complex
Integer
+--E
+
+--S 22 14:239 Schaums and Axiom agree
+hh:=complexNormalize gg
+--R
+--R (10) 0
+--R Type: Expression Complex
Integer
+--E
+
@
\section{\cite{1}:14.240~~~~~$\displaystyle
@@ -85,7 +312,7 @@ $$
<<*>>=
)clear all
---S 4 of 28
+--S 23
aa:=integrate(x^3/sqrt(a^2-x^2),x)
--R
--R
@@ -98,6 +325,27 @@ aa:=integrate(x^3/sqrt(a^2-x^2),x)
--R (3x - 12a )\|- x + a - 9a x + 12a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 24
+bb:=(a^2-x^2)^(3/2)/3-a^2*sqrt(a^2-x^2)
+--R
+--R +---------+
+--R 2 2 | 2 2
+--R (- x - 2a )\|- x + a
+--R (2) ------------------------
+--R 3
+--R Type: Expression
Integer
+--E
+
+--S 25 14:240 Schaums and Axiom differ by a constant
+cc:=aa-bb
+--R
+--R 3
+--R 2a
+--R (3) - ---
+--R 3
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.241~~~~~$\displaystyle\int{\frac{dx}{x\sqrt{a^2-x^2}}}$}
@@ -107,7 +355,7 @@ $$
<<*>>=
)clear all
---S 5 of 28
+--S 26
aa:=integrate(1/(x*sqrt(a^2-x^2)),x)
--R
--R
@@ -120,6 +368,66 @@ aa:=integrate(1/(x*sqrt(a^2-x^2)),x)
--R a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 27
+bb:=-1/a*log((a+sqrt(a^2-x^2))/x)
+--R
+--R +---------+
+--R | 2 2
+--R \|- x + a + a
+--R log(----------------)
+--R x
+--R (2) - ---------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 28
+cc:=aa-bb
+--R
+--R +---------+ +---------+
+--R | 2 2 | 2 2
+--R \|- x + a + a \|- x + a - a
+--R log(----------------) + log(----------------)
+--R x x
+--R (3) ---------------------------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 29
+dd:=expandLog cc
+--R
+--R +---------+ +---------+
+--R | 2 2 | 2 2
+--R log(\|- x + a + a) + log(\|- x + a - a) - 2log(x)
+--R (4) -------------------------------------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 30
+ee:=complexNormalize dd
+--R
+--R x
+--R 2log(-------)
+--R +----+
+--R | 2
+--R \|- x
+--R (5) - -------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 31 14:241 Schaums and Axiom differ by a constant
+ff:=rootSimp ee
+--R
+--R +---+
+--R 2log(\|- 1 )
+--R (6) ------------
+--R a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.242~~~~~$\displaystyle
@@ -130,7 +438,7 @@ $$
<<*>>=
)clear all
---S 6 of 28
+--S 32
aa:=integrate(1/(x^2*sqrt(a^2-x^2)),x)
--R
--R
@@ -143,6 +451,25 @@ aa:=integrate(1/(x^2*sqrt(a^2-x^2)),x)
--R a x\|- x + a - a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 33
+bb:=-sqrt(a^2-x^2)/(a^2*x)
+--R
+--R +---------+
+--R | 2 2
+--R \|- x + a
+--R (2) - ------------
+--R 2
+--R a x
+--R Type: Expression
Integer
+--E
+
+--S 34 14:242 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.243~~~~~$\displaystyle\int{\frac{dx}{x^3\sqrt{a^2-x^2}}}$}
@@ -153,7 +480,7 @@ $$
<<*>>=
)clear all
---S 7 of 28
+--S 35
aa:=integrate(1/(x^3*sqrt(a^2-x^2)),x)
--R
--R
@@ -173,6 +500,71 @@ aa:=integrate(1/(x^3*sqrt(a^2-x^2)),x)
--R 4a x \|- x + a + 2a x - 4a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 36
+bb:=-sqrt(a^2-x^2)/(2*a^2*x^2)-1/(2*a^3)*log((a+sqrt(a^2-x^2))/x)
+--R
+--R +---------+
+--R | 2 2 +---------+
+--R 2 \|- x + a + a | 2 2
+--R - x log(----------------) - a\|- x + a
+--R x
+--R (2) -----------------------------------------
+--R 3 2
+--R 2a x
+--R Type: Expression
Integer
+--E
+
+--S 37
+cc:=aa-bb
+--R
+--R +---------+ +---------+
+--R | 2 2 | 2 2
+--R \|- x + a + a \|- x + a - a
+--R log(----------------) + log(----------------)
+--R x x
+--R (3) ---------------------------------------------
+--R 3
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 38
+dd:=expandLog cc
+--R
+--R +---------+ +---------+
+--R | 2 2 | 2 2
+--R log(\|- x + a + a) + log(\|- x + a - a) - 2log(x)
+--R (4) -------------------------------------------------------
+--R 3
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 39
+ee:=complexNormalize dd
+--R
+--R x
+--R log(-------)
+--R +----+
+--R | 2
+--R \|- x
+--R (5) - ------------
+--R 3
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 40 14:243 Schaums and Axiom differ by a constant
+ff:=rootSimp ee
+--R
+--R +---+
+--R log(\|- 1 )
+--R (6) -----------
+--R 3
+--R a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.244~~~~~$\displaystyle\int{\sqrt{a^2-x^2}}~dx$}
@@ -182,7 +574,7 @@ $$
<<*>>=
)clear all
---S 8 of 28
+--S 41
aa:=integrate(sqrt(a^2-x^2),x)
--R
--R
@@ -202,6 +594,127 @@ aa:=integrate(sqrt(a^2-x^2),x)
--R 4a\|- x + a + 2x - 4a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 42
+bb:=(x*sqrt(a^2-x^2))/2+a^2/2*asin(x/a)
+--R
+--R +---------+
+--R | 2 2 2 x
+--R x\|- x + a + a asin(-)
+--R a
+--R (2) -------------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 43
+cc:=aa-bb
+--R
+--R +---------+
+--R | 2 2
+--R 2 \|- x + a - a 2 x
+--R - 2a atan(----------------) - a asin(-)
+--R x a
+--R (3) ---------------------------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 44
+asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2)))
+--R
+--R +--------+
+--R | 2
+--R (4) asin(x) == %i log(\|- x + 1 - %i x)
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 45
+dd:=asinrule cc
+--R
+--R +---------+
+--R | 2 2
+--R |- x + a
+--R a |--------- - %i x +---------+
+--R | 2 | 2 2
+--R 2 \| a 2 \|- x + a - a
+--R - %i a log(--------------------) - 2a atan(----------------)
+--R a x
+--R (5) ------------------------------------------------------------
+--R 2
+--R Type: Expression Complex
Integer
+--E
+
+--S 46
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R - x + %i
+--R %i log(--------)
+--R x + %i
+--R (6) atan(x) == - ----------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 47
+ee:=atanrule dd
+--R
+--R +---------+
+--R | 2 2
+--R |- x + a
+--R a |--------- - %i x +---------+
+--R | 2 | 2 2
+--R 2 \| a 2 - \|- x + a + %i x +
a
+--R - %i a log(--------------------) + %i a
log(-------------------------)
+--R a +---------+
+--R | 2 2
+--R \|- x + a + %i x - a
+--R (7)
----------------------------------------------------------------------
+--R 2
+--R Type: Expression Complex
Integer
+--E
+
+--S 48
+ff:=expandLog ee
+--R
+--R (8)
+--R +---------+
+--R | 2 2 +---------+
+--R 2 |- x + a 2 | 2 2
+--R - %i a log(a |--------- - %i x) - %i a log(\|- x + a + %i x - a)
+--R | 2
+--R \| a
+--R +
+--R +---------+
+--R 2 | 2 2 2 2
+--R %i a log(\|- x + a - %i x - a) + %i a log(a) + %i a log(- 1)
+--R /
+--R 2
+--R Type: Expression Complex
Integer
+--E
+
+--S 49
+gg:=rootSimp ff
+--R
+--R (9)
+--R +-------+ +-------+
+--R 2 | 2 2 2 | 2 2
+--R - %i a log(%i\|x - a + %i x - a) - %i a log(%i\|x - a - %i x)
+--R +
+--R +-------+
+--R 2 | 2 2 2 2
+--R %i a log(%i\|x - a - %i x - a) + %i a log(a) + %i a log(- 1)
+--R /
+--R 2
+--R Type: Expression Complex
Integer
+--E
+
+--S 50 14:244 Schaums and Axiom agree
+hh:=complexNormalize gg
+--R
+--R (10) 0
+--R Type: Expression Complex
Integer
+--E
@
\section{\cite{1}:14.245~~~~~$\displaystyle\int{x\sqrt{a^2-x^2}}~dx$}
@@ -211,7 +724,7 @@ $$
<<*>>=
)clear all
---S 9 of 28
+--S 51
aa:=integrate(x*sqrt(a^2-x^2),x)
--R
--R
@@ -224,6 +737,27 @@ aa:=integrate(x*sqrt(a^2-x^2),x)
--R (3x - 12a )\|- x + a - 9a x + 12a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 52
+bb:=-(a^2-x^2)^(3/2)/3
+--R
+--R +---------+
+--R 2 2 | 2 2
+--R (x - a )\|- x + a
+--R (2) ---------------------
+--R 3
+--R Type: Expression
Integer
+--E
+
+--S 53 14:245 Schaums and Axiom differ by a constant
+cc:=aa-bb
+--R
+--R 3
+--R a
+--R (3) - --
+--R 3
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.246~~~~~$\displaystyle
@@ -235,7 +769,7 @@ $$
<<*>>=
)clear all
---S 10 of 28
+--S 54
aa:=integrate(x^2*sqrt(a^2-x^2),x)
--R
--R
@@ -259,6 +793,126 @@ aa:=integrate(x^2*sqrt(a^2-x^2),x)
--R (32a x - 64a )\|- x + a + 8x - 64a x + 64a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 55
+bb:=-(x*(a^2-x^2)^(3/2))/4+(a^2*x*sqrt(a^2-x^2))/8+a^4/8*asin(x/a)
+--R
+--R +---------+
+--R 3 2 | 2 2 4 x
+--R (2x - a x)\|- x + a + a asin(-)
+--R a
+--R (2) -----------------------------------
+--R 8
+--R Type: Expression
Integer
+--E
+
+--S 56
+cc:=aa-bb
+--R
+--R +---------+
+--R | 2 2
+--R 4 \|- x + a - a 4 x
+--R - 2a atan(----------------) - a asin(-)
+--R x a
+--R (3) ---------------------------------------
+--R 8
+--R Type: Expression
Integer
+--E
+
+--S 57
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R - x + %i
+--R %i log(--------)
+--R x + %i
+--R (4) atan(x) == - ----------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 58
+dd:=atanrule cc
+--R
+--R +---------+
+--R | 2 2
+--R 4 - \|- x + a + %i x + a 4 x
+--R %i a log(-------------------------) - a asin(-)
+--R +---------+ a
+--R | 2 2
+--R \|- x + a + %i x - a
+--R (5) -----------------------------------------------
+--R 8
+--R Type: Expression Complex
Integer
+--E
+
+--S 59
+asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2)))
+--R
+--R +--------+
+--R | 2
+--R (6) asin(x) == %i log(\|- x + 1 - %i x)
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 60
+ee:=asinrule dd
+--R
+--R +---------+
+--R | 2 2
+--R |- x + a
+--R a |--------- - %i x +---------+
+--R | 2 | 2 2
+--R 4 \| a 4 - \|- x + a + %i x +
a
+--R - %i a log(--------------------) + %i a
log(-------------------------)
+--R a +---------+
+--R | 2 2
+--R \|- x + a + %i x - a
+--R (7)
----------------------------------------------------------------------
+--R 8
+--R Type: Expression Complex
Integer
+--E
+
+--S 61
+ff:=expandLog ee
+--R
+--R (8)
+--R +---------+
+--R | 2 2 +---------+
+--R 4 |- x + a 4 | 2 2
+--R - %i a log(a |--------- - %i x) - %i a log(\|- x + a + %i x - a)
+--R | 2
+--R \| a
+--R +
+--R +---------+
+--R 4 | 2 2 4 4
+--R %i a log(\|- x + a - %i x - a) + %i a log(a) + %i a log(- 1)
+--R /
+--R 8
+--R Type: Expression Complex
Integer
+--E
+
+--S 62
+gg:=rootSimp ff
+--R
+--R (9)
+--R +-------+ +-------+
+--R 4 | 2 2 4 | 2 2
+--R - %i a log(%i\|x - a + %i x - a) - %i a log(%i\|x - a - %i x)
+--R +
+--R +-------+
+--R 4 | 2 2 4 4
+--R %i a log(%i\|x - a - %i x - a) + %i a log(a) + %i a log(- 1)
+--R /
+--R 8
+--R Type: Expression Complex
Integer
+--E
+
+--S 63 14:246 Schaums and Axiom agree
+hh:=complexNormalize gg
+--R
+--R (10) 0
+--R Type: Expression Complex
Integer
+--E
@
\section{\cite{1}:14.247~~~~~$\displaystyle
@@ -269,7 +923,7 @@ $$
<<*>>=
)clear all
---S 11 of 28
+--S 64
aa:=integrate(x^3*sqrt(a^2-x^2),x)
--R
--R
@@ -283,6 +937,27 @@ aa:=integrate(x^3*sqrt(a^2-x^2),x)
--R (15x - 180a x + 240a )\|- x + a - 75a x + 300a x - 240a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 65
+bb:=(a^2-x^2)^(5/2)/5-(a^2*(a^2-x^2)^(3/2))/3
+--R
+--R +---------+
+--R 4 2 2 4 | 2 2
+--R (3x - a x - 2a )\|- x + a
+--R (2) ------------------------------
+--R 15
+--R Type: Expression
Integer
+--E
+
+--S 66 14:247 Schaums and Axiom differ by a constant
+cc:=aa-bb
+--R
+--R 5
+--R 2a
+--R (3) - ---
+--R 15
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.248~~~~~$\displaystyle
@@ -293,7 +968,7 @@ $$
<<*>>=
)clear all
---S 12 of 28
+--S 67
aa:=integrate(sqrt(a^2-x^2)/x,x)
--R
--R
@@ -308,6 +983,56 @@ aa:=integrate(sqrt(a^2-x^2)/x,x)
--R \|- x + a - a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 68
+bb:=sqrt(a^2-x^2)-a*log((a+sqrt(a^2-x^2))/x)
+--R
+--R +---------+
+--R | 2 2 +---------+
+--R \|- x + a + a | 2 2
+--R (2) - a log(----------------) + \|- x + a
+--R x
+--R Type: Expression
Integer
+--E
+
+--S 69
+cc:=aa-bb
+--R
+--R +---------+ +---------+
+--R | 2 2 | 2 2
+--R \|- x + a + a \|- x + a - a
+--R (3) a log(----------------) + a log(----------------) + a
+--R x x
+--R Type: Expression
Integer
+--E
+
+--S 70
+dd:=expandLog cc
+--R
+--R +---------+ +---------+
+--R | 2 2 | 2 2
+--R (4) a log(\|- x + a + a) + a log(\|- x + a - a) - 2a log(x) + a
+--R Type: Expression
Integer
+--E
+
+--S 71
+ee:=complexNormalize dd
+--R
+--R x
+--R (5) - 2a log(-------) + a
+--R +----+
+--R | 2
+--R \|- x
+--R Type: Expression
Integer
+--E
+
+--S 72 14:248 Schaums and Axiom differ by a constant
+ff:=rootSimp ee
+--R
+--R +---+
+--R (6) 2a log(\|- 1 ) + a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.249~~~~~$\displaystyle
@@ -318,7 +1043,7 @@ $$
<<*>>=
)clear all
---S 13 of 28
+--S 73
aa:=integrate(sqrt(a^2-x^2)/x^2,x)
--R
--R
@@ -334,6 +1059,117 @@ aa:=integrate(sqrt(a^2-x^2)/x^2,x)
--R x\|- x + a - a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 74
+bb:=-sqrt(a^2-x^2)/x-asin(x/a)
+--R
+--R +---------+
+--R | 2 2 x
+--R - \|- x + a - x asin(-)
+--R a
+--R (2) --------------------------
+--R x
+--R Type: Expression
Integer
+--E
+
+--S 75
+cc:=aa-bb
+--R
+--R +---------+
+--R | 2 2
+--R \|- x + a - a x
+--R (3) 2atan(----------------) + asin(-)
+--R x a
+--R Type: Expression
Integer
+--E
+
+--S 76
+asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2)))
+--R
+--R +--------+
+--R | 2
+--R (4) asin(x) == %i log(\|- x + 1 - %i x)
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 77
+dd:=asinrule cc
+--R
+--R +---------+
+--R | 2 2
+--R |- x + a
+--R a |--------- - %i x +---------+
+--R | 2 | 2 2
+--R \| a \|- x + a - a
+--R (5) %i log(--------------------) + 2atan(----------------)
+--R a x
+--R Type: Expression Complex
Integer
+--E
+
+--S 78
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R - x + %i
+--R %i log(--------)
+--R x + %i
+--R (6) atan(x) == - ----------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 79
+ee:=atanrule dd
+--R
+--R +---------+
+--R | 2 2
+--R |- x + a
+--R a |--------- - %i x +---------+
+--R | 2 | 2 2
+--R \| a - \|- x + a + %i x + a
+--R (7) %i log(--------------------) - %i log(-------------------------)
+--R a +---------+
+--R | 2 2
+--R \|- x + a + %i x - a
+--R Type: Expression Complex
Integer
+--E
+
+--S 80
+ff:=expandLog ee
+--R
+--R (8)
+--R +---------+
+--R | 2 2 +---------+
+--R |- x + a | 2 2
+--R %i log(a |--------- - %i x) + %i log(\|- x + a + %i x - a)
+--R | 2
+--R \| a
+--R +
+--R +---------+
+--R | 2 2
+--R - %i log(\|- x + a - %i x - a) - %i log(a) - %i log(- 1)
+--R Type: Expression Complex
Integer
+--E
+
+--S 81
+gg:=rootSimp ff
+--R
+--R (9)
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R %i log(%i\|x - a + %i x - a) + %i log(%i\|x - a - %i x)
+--R +
+--R +-------+
+--R | 2 2
+--R - %i log(%i\|x - a - %i x - a) - %i log(a) - %i log(- 1)
+--R Type: Expression Complex
Integer
+--E
+
+--S 82 14:249 Schaums and Axiom agree
+hh:=complexNormalize gg
+--R
+--R (10) 0
+--R Type: Expression Complex
Integer
+--E
@
\section{\cite{1}:14.250~~~~~$\displaystyle
@@ -345,7 +1181,7 @@ $$
<<*>>=
)clear all
---S 14 of 28
+--S 83
aa:=integrate(sqrt(a^2-x^2)/x^3,x)
--R
--R
@@ -365,6 +1201,67 @@ aa:=integrate(sqrt(a^2-x^2)/x^3,x)
--R 4a x \|- x + a + 2a x - 4a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 84
+bb:=-sqrt(a^2-x^2)/(2*x^2)+1/(2*a)*log((a+sqrt(a^2-x^2))/x)
+--R
+--R +---------+
+--R | 2 2 +---------+
+--R 2 \|- x + a + a | 2 2
+--R x log(----------------) - a\|- x + a
+--R x
+--R (2) ---------------------------------------
+--R 2
+--R 2a x
+--R Type: Expression
Integer
+--E
+
+--S 85
+cc:=aa-bb
+--R
+--R +---------+ +---------+
+--R | 2 2 | 2 2
+--R \|- x + a + a \|- x + a - a
+--R - log(----------------) - log(----------------)
+--R x x
+--R (3) -----------------------------------------------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 86
+dd:=expandLog cc
+--R
+--R +---------+ +---------+
+--R | 2 2 | 2 2
+--R - log(\|- x + a + a) - log(\|- x + a - a) + 2log(x)
+--R (4) ---------------------------------------------------------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 87
+ee:=complexNormalize dd
+--R
+--R x
+--R log(-------)
+--R +----+
+--R | 2
+--R \|- x
+--R (5) ------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 88 14:250 Schaums and Axiom differ by a constant
+ff:=rootSimp ee
+--R
+--R +---+
+--R log(\|- 1 )
+--R (6) - -----------
+--R a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.251~~~~~$\displaystyle\int{\frac{dx}{(a^2-x^2)^{3/2}}}$}
@@ -374,7 +1271,7 @@ $$
<<*>>=
)clear all
---S 15 of 28
+--S 89
aa:=integrate(1/(a^2-x^2)^(3/2),x)
--R
--R
@@ -387,6 +1284,24 @@ aa:=integrate(1/(a^2-x^2)^(3/2),x)
--R a \|- x + a + a x - a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 90
+bb:=x/(a^2*sqrt(a^2-x^2))
+--R
+--R x
+--R (2) --------------
+--R +---------+
+--R 2 | 2 2
+--R a \|- x + a
+--R Type: Expression
Integer
+--E
+
+--S 91 14:251 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.252~~~~~$\displaystyle
@@ -397,7 +1312,7 @@ $$
<<*>>=
)clear all
---S 16 of 28
+--S 92
aa:=integrate(x/(a^2-x^2)^(3/2),x)
--R
--R
@@ -409,6 +1324,26 @@ aa:=integrate(x/(a^2-x^2)^(3/2),x)
--R a \|- x + a + a x - a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 93
+bb:=1/sqrt(a^2-x^2)
+--R
+--R 1
+--R (2) ------------
+--R +---------+
+--R | 2 2
+--R \|- x + a
+--R Type: Expression
Integer
+--E
+
+--S 94 14:252 Schaums and Axiom differ by a constant
+cc:=aa-bb
+--R
+--R 1
+--R (3) -
+--R a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.253~~~~~$\displaystyle
@@ -419,7 +1354,7 @@ $$
<<*>>=
)clear all
---S 17 of 28
+--S 95
aa:=integrate(x^2/(a^2-x^2)^(3/2),x)
--R
--R
@@ -435,6 +1370,118 @@ aa:=integrate(x^2/(a^2-x^2)^(3/2),x)
--R a\|- x + a + x - a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 96
+bb:=x/sqrt(a^2-x^2)-asin(x/a)
+--R
+--R +---------+
+--R x | 2 2
+--R - asin(-)\|- x + a + x
+--R a
+--R (2) -------------------------
+--R +---------+
+--R | 2 2
+--R \|- x + a
+--R Type: Expression
Integer
+--E
+
+--S 97
+cc:=aa-bb
+--R
+--R +---------+
+--R | 2 2
+--R \|- x + a - a x
+--R (3) 2atan(----------------) + asin(-)
+--R x a
+--R Type: Expression
Integer
+--E
+
+--S 98
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R - x + %i
+--R %i log(--------)
+--R x + %i
+--R (4) atan(x) == - ----------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 99
+dd:=atanrule cc
+--R
+--R +---------+
+--R | 2 2
+--R - \|- x + a + %i x + a x
+--R (5) - %i log(-------------------------) + asin(-)
+--R +---------+ a
+--R | 2 2
+--R \|- x + a + %i x - a
+--R Type: Expression Complex
Integer
+--E
+
+--S 100
+asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2)))
+--R
+--R +--------+
+--R | 2
+--R (6) asin(x) == %i log(\|- x + 1 - %i x)
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 101
+ee:=asinrule dd
+--R
+--R +---------+
+--R | 2 2
+--R |- x + a
+--R a |--------- - %i x +---------+
+--R | 2 | 2 2
+--R \| a - \|- x + a + %i x + a
+--R (7) %i log(--------------------) - %i log(-------------------------)
+--R a +---------+
+--R | 2 2
+--R \|- x + a + %i x - a
+--R Type: Expression Complex
Integer
+--E
+
+--S 102
+ff:=expandLog ee
+--R
+--R (8)
+--R +---------+
+--R | 2 2 +---------+
+--R |- x + a | 2 2
+--R %i log(a |--------- - %i x) + %i log(\|- x + a + %i x - a)
+--R | 2
+--R \| a
+--R +
+--R +---------+
+--R | 2 2
+--R - %i log(\|- x + a - %i x - a) - %i log(a) - %i log(- 1)
+--R Type: Expression Complex
Integer
+--E
+
+--S 103
+gg:=rootSimp ff
+--R
+--R (9)
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R %i log(%i\|x - a + %i x - a) + %i log(%i\|x - a - %i x)
+--R +
+--R +-------+
+--R | 2 2
+--R - %i log(%i\|x - a - %i x - a) - %i log(a) - %i log(- 1)
+--R Type: Expression Complex
Integer
+--E
+
+--S 104 14:253 Schaums and Axiom agree
+hh:=complexNormalize gg
+--R
+--R (10) 0
+--R Type: Expression Complex
Integer
+--E
@
\section{\cite{1}:14.254~~~~~$\displaystyle
@@ -445,7 +1492,7 @@ $$
<<*>>=
)clear all
---S 18 of 28
+--S 105
aa:=integrate(x^3/(a^2-x^2)^(3/2),x)
--R
--R
@@ -457,6 +1504,25 @@ aa:=integrate(x^3/(a^2-x^2)^(3/2),x)
--R (x - 2a )\|- x + a - 2a x + 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 106
+bb:=sqrt(a^2-x^2)+a^2/sqrt(a^2-x^2)
+--R
+--R 2 2
+--R - x + 2a
+--R (2) ------------
+--R +---------+
+--R | 2 2
+--R \|- x + a
+--R Type: Expression
Integer
+--E
+
+--S 107 14:254 Schaums and Axiom differ by a constant
+cc:=aa-bb
+--R
+--R (3) 2a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.255~~~~~$\displaystyle
@@ -468,7 +1534,7 @@ $$
<<*>>=
)clear all
---S 19 of 28
+--S 108
aa:=integrate(1/(x*(a^2-x^2)^(3/2)),x)
--R
--R
@@ -483,6 +1549,72 @@ aa:=integrate(1/(x*(a^2-x^2)^(3/2)),x)
--R a \|- x + a + a x - a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 109
+bb:=1/(a^2*sqrt(a^2-x^2))-1/a^3*log((a+sqrt(a^2-x^2))/x)
+--R
+--R +---------+
+--R +---------+ | 2 2
+--R | 2 2 \|- x + a + a
+--R - \|- x + a log(----------------) + a
+--R x
+--R (2) ---------------------------------------
+--R +---------+
+--R 3 | 2 2
+--R a \|- x + a
+--R Type: Expression
Integer
+--E
+
+--S 110
+cc:=aa-bb
+--R
+--R +---------+ +---------+
+--R | 2 2 | 2 2
+--R \|- x + a + a \|- x + a - a
+--R log(----------------) + log(----------------) + 1
+--R x x
+--R (3) -------------------------------------------------
+--R 3
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 111
+dd:=expandLog cc
+--R
+--R +---------+ +---------+
+--R | 2 2 | 2 2
+--R log(\|- x + a + a) + log(\|- x + a - a) - 2log(x) + 1
+--R (4) -----------------------------------------------------------
+--R 3
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 112
+ee:=complexNormalize dd
+--R
+--R x
+--R - 2log(-------) + 1
+--R +----+
+--R | 2
+--R \|- x
+--R (5) -------------------
+--R 3
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 113 14:255 Schaums and Axiom differ by a constant
+ff:=rootSimp ee
+--R
+--R +---+
+--R 2log(\|- 1 ) + 1
+--R (6) ----------------
+--R 3
+--R a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.256~~~~~$\displaystyle
@@ -493,7 +1625,7 @@ $$
<<*>>=
)clear all
---S 20 of 28
+--S 114
aa:=integrate(1/(x^2*(a^2-x^2)^(3/2)),x)
--R
--R
@@ -506,6 +1638,25 @@ aa:=integrate(1/(x^2*(a^2-x^2)^(3/2)),x)
--R (a x - 2a x)\|- x + a - 2a x + 2a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 115
+bb:=-sqrt(a^2-x^2)/(a^4*x)+x/(a^4*sqrt(a^2-x^2))
+--R
+--R 2 2
+--R 2x - a
+--R (2) ---------------
+--R +---------+
+--R 4 | 2 2
+--R a x\|- x + a
+--R Type: Expression
Integer
+--E
+
+--S 116 14:256 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.257~~~~~$\displaystyle
@@ -518,7 +1669,7 @@ $$
<<*>>=
)clear all
---S 21 of 28
+--S 117
aa:=integrate(1/(x^3*(a^2-x^2)^(3/2)),x)
--R
--R
@@ -542,6 +1693,72 @@ aa:=integrate(1/(x^3*(a^2-x^2)^(3/2)),x)
--R (6a x - 8a x )\|- x + a + 2a x - 10a x + 8a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 118
+bb:=-1/(2*a^2*x^2*sqrt(a^2-x^2))+3/(2*a^4*sqrt(a^2-x^2))-3/(2*a^5)*log((a+sqrt(a^2-x^2))/x)
+--R
+--R +---------+
+--R +---------+ | 2 2
+--R 2 | 2 2 \|- x + a + a 2 3
+--R - 3x \|- x + a log(----------------) + 3a x - a
+--R x
+--R (2) ---------------------------------------------------
+--R +---------+
+--R 5 2 | 2 2
+--R 2a x \|- x + a
+--R Type: Expression
Integer
+--E
+
+--S 119
+cc:=aa-bb
+--R
+--R +---------+ +---------+
+--R | 2 2 | 2 2
+--R \|- x + a + a \|- x + a - a
+--R 3log(----------------) + 3log(----------------) + 2
+--R x x
+--R (3) ---------------------------------------------------
+--R 5
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 120
+dd:=expandLog cc
+--R
+--R +---------+ +---------+
+--R | 2 2 | 2 2
+--R 3log(\|- x + a + a) + 3log(\|- x + a - a) - 6log(x) + 2
+--R (4) -------------------------------------------------------------
+--R 5
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 121
+ee:=complexNormalize dd
+--R
+--R x
+--R - 3log(-------) + 1
+--R +----+
+--R | 2
+--R \|- x
+--R (5) -------------------
+--R 5
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 122 14:257 Schaums and Axiom differ by a constant
+ff:=rootSimp ee
+--R
+--R +---+
+--R 3log(\|- 1 ) + 1
+--R (6) ----------------
+--R 5
+--R a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.258~~~~~$\displaystyle\int{(a^2-x^2)^{3/2}}~dx$}
@@ -552,7 +1769,7 @@ $$
<<*>>=
)clear all
---S 22 of 28
+--S 123
aa:=integrate((a^2-x^2)^(3/2),x)
--R
--R
@@ -579,6 +1796,128 @@ aa:=integrate((a^2-x^2)^(3/2),x)
--R (32a x - 64a )\|- x + a + 8x - 64a x + 64a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 124
+bb:=(x*(a^2-x^2)^(3/2))/4+(3*a^2*x*sqrt(a^2-x^2))/8+3/8*a^4*asin(x/a)
+--R
+--R +---------+
+--R 3 2 | 2 2 4 x
+--R (- 2x + 5a x)\|- x + a + 3a asin(-)
+--R a
+--R (2) ---------------------------------------
+--R 8
+--R Type: Expression
Integer
+--E
+
+--S 125
+cc:=aa-bb
+--R
+--R +---------+
+--R | 2 2
+--R 4 \|- x + a - a 4 x
+--R - 6a atan(----------------) - 3a asin(-)
+--R x a
+--R (3) ----------------------------------------
+--R 8
+--R Type: Expression
Integer
+--E
+
+--S 126
+asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2)))
+--R
+--R +--------+
+--R | 2
+--R (4) asin(x) == %i log(\|- x + 1 - %i x)
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 127
+ee:=asinrule cc
+--R
+--R +---------+
+--R | 2 2
+--R |- x + a
+--R a |--------- - %i x +---------+
+--R | 2 | 2 2
+--R 4 \| a 4 \|- x + a - a
+--R - 3%i a log(--------------------) - 6a atan(----------------)
+--R a x
+--R (5) -------------------------------------------------------------
+--R 8
+--R Type: Expression Complex
Integer
+--E
+
+--S 128
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R - x + %i
+--R %i log(--------)
+--R x + %i
+--R (6) atan(x) == - ----------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 129
+ff:=atanrule ee
+--R
+--R (7)
+--R +---------+
+--R | 2 2
+--R |- x + a
+--R a |--------- - %i x +---------+
+--R | 2 | 2 2
+--R 4 \| a 4 - \|- x + a + %i x + a
+--R - 3%i a log(--------------------) + 3%i a log(-------------------------)
+--R a +---------+
+--R | 2 2
+--R \|- x + a + %i x - a
+--R ------------------------------------------------------------------------
+--R 8
+--R Type: Expression Complex
Integer
+--E
+
+--S 130
+gg:=expandLog ff
+--R
+--R (8)
+--R +---------+
+--R | 2 2 +---------+
+--R 4 |- x + a 4 | 2 2
+--R - 3%i a log(a |--------- - %i x) - 3%i a log(\|- x + a + %i x -
a)
+--R | 2
+--R \| a
+--R +
+--R +---------+
+--R 4 | 2 2 4 4
+--R 3%i a log(\|- x + a - %i x - a) + 3%i a log(a) + 3%i a log(- 1)
+--R /
+--R 8
+--R Type: Expression Complex
Integer
+--E
+
+--S 131
+hh:=rootSimp gg
+--R
+--R (9)
+--R +-------+ +-------+
+--R 4 | 2 2 4 | 2 2
+--R - 3%i a log(%i\|x - a + %i x - a) - 3%i a log(%i\|x - a - %i x)
+--R +
+--R +-------+
+--R 4 | 2 2 4 4
+--R 3%i a log(%i\|x - a - %i x - a) + 3%i a log(a) + 3%i a log(- 1)
+--R /
+--R 8
+--R Type: Expression Complex
Integer
+--E
+
+--S 132 14:258 Schaums and Axiom agree
+ii:=complexNormalize hh
+--R
+--R (10) 0
+--R Type: Expression Complex
Integer
+--E
@
\section{\cite{1}:14.259~~~~~$\displaystyle\int{x(a^2-x^2)^{3/2}}~dx$}
@@ -586,7 +1925,7 @@ $$\int{x(a^2-x^2)^{3/2}}=\frac{(a^2-x^2)^{5/2}}{5}$$
<<*>>=
)clear all
---S 23 of 28
+--S 133
aa:=integrate(x*(a^2-x^2)^(3/2),x)
--R
--R
@@ -603,6 +1942,27 @@ aa:=integrate(x*(a^2-x^2)^(3/2),x)
--R (5x - 60a x + 80a )\|- x + a - 25a x + 100a x - 80a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 134
+bb:=-(a^2-x^2)^(5/2)/5
+--R
+--R +---------+
+--R 4 2 2 4 | 2 2
+--R (- x + 2a x - a )\|- x + a
+--R (2) -------------------------------
+--R 5
+--R Type: Expression
Integer
+--E
+
+--S 135 14:259 Schaums and Axiom differ by a constant
+cc:=aa-bb
+--R
+--R 5
+--R a
+--R (3) - --
+--R 5
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.260~~~~~$\displaystyle\int{x^2(a^2-x^2)^{3/2}}~dx$}
@@ -614,7 +1974,7 @@ $$
<<*>>=
)clear all
---S 24 of 28
+--S 136
aa:=integrate(x^2*(a^2-x^2)^(3/2),x)
--R
--R
@@ -647,6 +2007,126 @@ aa:=integrate(x^2*(a^2-x^2)^(3/2),x)
--R - 1536a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 137
+bb:=-(x*(a^2-x^2)^(5/2))/6+(a^2*x*(a^2-x^2)^(3/2))/24+(a^4*x*sqrt(a^2-x^2))/16+a^6/16*asin(x/a)
+--R
+--R +---------+
+--R 5 2 3 4 | 2 2 6 x
+--R (- 8x + 14a x - 3a x)\|- x + a + 3a asin(-)
+--R a
+--R (2) ------------------------------------------------
+--R 48
+--R Type: Expression
Integer
+--E
+
+--S 138
+cc:=aa-bb
+--R
+--R +---------+
+--R | 2 2
+--R 6 \|- x + a - a 6 x
+--R - 2a atan(----------------) - a asin(-)
+--R x a
+--R (3) ---------------------------------------
+--R 16
+--R Type: Expression
Integer
+--E
+
+--S 139
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R - x + %i
+--R %i log(--------)
+--R x + %i
+--R (4) atan(x) == - ----------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 140
+dd:=atanrule cc
+--R
+--R +---------+
+--R | 2 2
+--R 6 - \|- x + a + %i x + a 6 x
+--R %i a log(-------------------------) - a asin(-)
+--R +---------+ a
+--R | 2 2
+--R \|- x + a + %i x - a
+--R (5) -----------------------------------------------
+--R 16
+--R Type: Expression Complex
Integer
+--E
+
+--S 141
+asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2)))
+--R
+--R +--------+
+--R | 2
+--R (6) asin(x) == %i log(\|- x + 1 - %i x)
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 142
+ee:=asinrule dd
+--R
+--R +---------+
+--R | 2 2
+--R |- x + a
+--R a |--------- - %i x +---------+
+--R | 2 | 2 2
+--R 6 \| a 6 - \|- x + a + %i x +
a
+--R - %i a log(--------------------) + %i a
log(-------------------------)
+--R a +---------+
+--R | 2 2
+--R \|- x + a + %i x - a
+--R (7)
----------------------------------------------------------------------
+--R 16
+--R Type: Expression Complex
Integer
+--E
+
+--S 143
+ff:=expandLog ee
+--R
+--R (8)
+--R +---------+
+--R | 2 2 +---------+
+--R 6 |- x + a 6 | 2 2
+--R - %i a log(a |--------- - %i x) - %i a log(\|- x + a + %i x - a)
+--R | 2
+--R \| a
+--R +
+--R +---------+
+--R 6 | 2 2 6 6
+--R %i a log(\|- x + a - %i x - a) + %i a log(a) + %i a log(- 1)
+--R /
+--R 16
+--R Type: Expression Complex
Integer
+--E
+
+--S 144
+gg:=rootSimp ff
+--R
+--R (9)
+--R +-------+ +-------+
+--R 6 | 2 2 6 | 2 2
+--R - %i a log(%i\|x - a + %i x - a) - %i a log(%i\|x - a - %i x)
+--R +
+--R +-------+
+--R 6 | 2 2 6 6
+--R %i a log(%i\|x - a - %i x - a) + %i a log(a) + %i a log(- 1)
+--R /
+--R 16
+--R Type: Expression Complex
Integer
+--E
+
+--S 145 14:260 Schaums and Axiom agree
+hh:=complexNormalize gg
+--R
+--R (10) 0
+--R Type: Expression Complex
Integer
+--E
@
\section{\cite{1}:14.261~~~~~$\displaystyle\int{x^3(a^2-x^2)^{3/2}}~dx$}
@@ -656,7 +2136,7 @@ $$
<<*>>=
)clear all
---S 25 of 28
+--S 146
aa:=integrate(x^3*(a^2-x^2)^(3/2),x)
--R
--R
@@ -676,6 +2156,27 @@ aa:=integrate(x^3*(a^2-x^2)^(3/2),x)
--R - 3920a x + 2240a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 147
+bb:=(a^2-x^2)^(7/2)/7-(a^2*(a^2-x^2)^(5/2))/5
+--R
+--R +---------+
+--R 6 2 4 4 2 6 | 2 2
+--R (- 5x + 8a x - a x - 2a )\|- x + a
+--R (2) ----------------------------------------
+--R 35
+--R Type: Expression
Integer
+--E
+
+--S 148 14:261 Schaums and Axiom differ by a constant
+cc:=aa-bb
+--R
+--R 7
+--R 2a
+--R (3) - ---
+--R 35
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.262~~~~~$\displaystyle
@@ -687,7 +2188,7 @@ $$
<<*>>=
)clear all
---S 26 of 28
+--S 149
aa:=integrate((a^2-x^2)^(3/2)/x,x)
--R
--R
@@ -707,6 +2208,66 @@ aa:=integrate((a^2-x^2)^(3/2)/x,x)
--R (3x - 12a )\|- x + a - 9a x + 12a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 150
+bb:=(a^2-x^2)^(3/2)/3+a^2*sqrt(a^2-x^2)-a^3*log((a+sqrt(a^2-x^2))/x)
+--R
+--R +---------+
+--R | 2 2 +---------+
+--R 3 \|- x + a + a 2 2 | 2 2
+--R - 3a log(----------------) + (- x + 4a )\|- x + a
+--R x
+--R (2) -----------------------------------------------------
+--R 3
+--R Type: Expression
Integer
+--E
+
+--S 151
+cc:=aa-bb
+--R
+--R +---------+ +---------+
+--R | 2 2 | 2 2
+--R 3 \|- x + a + a 3 \|- x + a - a 3
+--R 3a log(----------------) + 3a log(----------------) + 4a
+--R x x
+--R (3) ---------------------------------------------------------
+--R 3
+--R Type: Expression
Integer
+--E
+
+--S 152
+dd:=expandLog cc
+--R
+--R +---------+ +---------+
+--R 3 | 2 2 3 | 2 2 3
3
+--R 3a log(\|- x + a + a) + 3a log(\|- x + a - a) - 6a log(x) + 4a
+--R (4)
---------------------------------------------------------------------
+--R 3
+--R Type: Expression
Integer
+--E
+
+--S 153
+ee:=complexNormalize dd
+--R
+--R 3 x 3
+--R - 6a log(-------) + 4a
+--R +----+
+--R | 2
+--R \|- x
+--R (5) -----------------------
+--R 3
+--R Type: Expression
Integer
+--E
+
+--S 154 14:262 Schaums and Axiom differ by a constant
+ff:=rootSimp ee
+--R
+--R 3 +---+ 3
+--R 6a log(\|- 1 ) + 4a
+--R (6) --------------------
+--R 3
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.263~~~~~$\displaystyle
@@ -718,7 +2279,7 @@ $$
<<*>>=
)clear all
---S 27 of 28
+--S 155
aa:=integrate((a^2-x^2)^{3/2}/x^2,x)
--R
--R
@@ -738,6 +2299,127 @@ aa:=integrate((a^2-x^2)^{3/2}/x^2,x)
--R (2x - 8a x)\|- x + a - 6a x + 8a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 156
+bb:=-(a^2-x^2)^(3/2)/x-(3*x*sqrt(a^2-x^2))/2-3/2*a^2*asin(x/a)
+--R
+--R +---------+
+--R 2 2 | 2 2 2 x
+--R (- x - 2a )\|- x + a - 3a x asin(-)
+--R a
+--R (2) ---------------------------------------
+--R 2x
+--R Type: Expression
Integer
+--E
+
+--S 157
+cc:=aa-bb
+--R
+--R +---------+
+--R | 2 2
+--R 2 \|- x + a - a 2 x
+--R 6a atan(----------------) + 3a asin(-)
+--R x a
+--R (3) --------------------------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 158
+asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2)))
+--R
+--R +--------+
+--R | 2
+--R (4) asin(x) == %i log(\|- x + 1 - %i x)
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 159
+dd:=asinrule cc
+--R
+--R +---------+
+--R | 2 2
+--R |- x + a
+--R a |--------- - %i x +---------+
+--R | 2 | 2 2
+--R 2 \| a 2 \|- x + a - a
+--R 3%i a log(--------------------) + 6a atan(----------------)
+--R a x
+--R (5) -----------------------------------------------------------
+--R 2
+--R Type: Expression Complex
Integer
+--E
+
+--S 160
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R - x + %i
+--R %i log(--------)
+--R x + %i
+--R (6) atan(x) == - ----------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 161
+ee:=atanrule dd
+--R
+--R +---------+
+--R | 2 2
+--R |- x + a
+--R a |--------- - %i x +---------+
+--R | 2 | 2 2
+--R 2 \| a 2 - \|- x + a + %i x +
a
+--R 3%i a log(--------------------) - 3%i a
log(-------------------------)
+--R a +---------+
+--R | 2 2
+--R \|- x + a + %i x - a
+--R (7)
----------------------------------------------------------------------
+--R 2
+--R Type: Expression Complex
Integer
+--E
+
+--S 162
+ff:=expandLog ee
+--R
+--R (8)
+--R +---------+
+--R | 2 2 +---------+
+--R 2 |- x + a 2 | 2 2
+--R 3%i a log(a |--------- - %i x) + 3%i a log(\|- x + a + %i x - a)
+--R | 2
+--R \| a
+--R +
+--R +---------+
+--R 2 | 2 2 2 2
+--R - 3%i a log(\|- x + a - %i x - a) - 3%i a log(a) - 3%i a log(- 1)
+--R /
+--R 2
+--R Type: Expression Complex
Integer
+--E
+
+--S 163
+gg:=rootSimp ff
+--R
+--R (9)
+--R +-------+ +-------+
+--R 2 | 2 2 2 | 2 2
+--R 3%i a log(%i\|x - a + %i x - a) + 3%i a log(%i\|x - a - %i x)
+--R +
+--R +-------+
+--R 2 | 2 2 2 2
+--R - 3%i a log(%i\|x - a - %i x - a) - 3%i a log(a) - 3%i a log(- 1)
+--R /
+--R 2
+--R Type: Expression Complex
Integer
+--E
+
+--S 164 14:263 Schaums and Axiom agree
+hh:=complexNormalize gg
+--R
+--R (10) 0
+--R Type: Expression Complex
Integer
+--E
@
\section{\cite{1}:14.264~~~~~$\displaystyle
@@ -749,7 +2431,7 @@ $$
<<*>>=
)clear all
---S 28 of 28
+--S 165
aa:=integrate((a^2-x^2)^(3/2)/x^3,x)
--R
--R
@@ -770,6 +2452,63 @@ aa:=integrate((a^2-x^2)^(3/2)/x^3,x)
--R Type: Union(Expression
Integer,...)
--E
+--S 166
+bb:=-(a^2-x^2)^(3/2)/(2*x^2)-(3*sqrt(a^2-x^2))/2+3/2*a*log((a+sqrt(a^2-x^2))/x)
+--R
+--R +---------+
+--R | 2 2 +---------+
+--R 2 \|- x + a + a 2 2 | 2 2
+--R 3a x log(----------------) + (- 2x - a )\|- x + a
+--R x
+--R (2) -----------------------------------------------------
+--R 2
+--R 2x
+--R Type: Expression
Integer
+--E
+
+--S 167
+cc:=aa-bb
+--R
+--R +---------+ +---------+
+--R | 2 2 | 2 2
+--R \|- x + a + a \|- x + a - a
+--R - 3a log(----------------) - 3a log(----------------) - 2a
+--R x x
+--R (3) ----------------------------------------------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 168
+dd:=expandLog cc
+--R
+--R +---------+ +---------+
+--R | 2 2 | 2 2
+--R - 3a log(\|- x + a + a) - 3a log(\|- x + a - a) + 6a log(x) -
2a
+--R (4)
----------------------------------------------------------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 169
+ee:=complexNormalize dd
+--R
+--R x
+--R (5) 3a log(-------) - a
+--R +----+
+--R | 2
+--R \|- x
+--R Type: Expression
Integer
+--E
+
+--S 170 14:264 Schaums and Axiom differ by a constant
+ff:=rootSimp ee
+--R
+--R +---+
+--R (6) - 3a log(\|- 1 ) - a
+--R Type: Expression
Integer
+--E
+
)spool
)lisp (bye)
@
diff --git a/src/input/schaum2.input.pamphlet b/src/input/schaum2.input.pamphlet
index 184122e..dfbaf88 100644
--- a/src/input/schaum2.input.pamphlet
+++ b/src/input/schaum2.input.pamphlet
@@ -742,58 +742,11 @@ cc:=aa-bb
--R 15a
--R Type: Expression
Integer
--E
-@
-If we had the terms
-<<*>>=
---S 52
-t1:=(3*a*x-2*b)
---R
---R (4) 3a x - 2b
---R Type: Polynomial
Integer
---E
---S 53
-t2:=(a*x+b)
---R
---R (5) a x + b
---R Type: Polynomial
Integer
---E
-@
-We can construct the Axiom result
-<<*>>=
---S 54
-2*t1*t2*sqrt(t2)/(15*a^2)
---R
---R 2 2 2 +-------+
---R (6a x + 2a b x - 4b )\|a x + b
---R (6) --------------------------------
---R 2
---R 15a
---R Type: Expression
Integer
---E
-@
-and we can construct the Spiegel result
-<<*>>=
---S 55
-2*t1*sqrt(t2^3)/(15*a^2)
+--S 52 14:90 Schaums and Axiom agree
+dd:=rootSimp cc
--R
---R +----------------------------+
---R | 3 3 2 2 2 3
---R (6a x - 4b)\|a x + 3a b x + 3a b x + b
---R (7) ------------------------------------------
---R 2
---R 15a
---R Type: Expression
Integer
---E
-@
-the difference of these two depends on
-<<*>>=
---S 56 14:90 Axiom cannot simplify this expression
-t2*sqrt(t2)-sqrt(t2^3)
---R
---R +----------------------------+
---R | 3 3 2 2 2 3 +-------+
---R (8) - \|a x + 3a b x + 3a b x + b + (a x + b)\|a x + b
+--R (4) 0
--R Type: Expression
Integer
--E
@
@@ -807,7 +760,7 @@ Note: the sqrt term is almost certainly $\sqrt{(ax+b)}$
<<*>>=
)clear all
---S 57
+--S 53
aa:=integrate(x^2*sqrt(a*x+b),x)
--R
--R
@@ -819,27 +772,26 @@ aa:=integrate(x^2*sqrt(a*x+b),x)
--R Type: Union(Expression
Integer,...)
--E
---S 58
-bb:=(2*(15*a^2*x^2-12*a*b*x+8*b^2))/(105*a^2)*sqrt((a*x+b)^3)
+--S 54
+bb:=(2*(15*a^2*x^2-12*a*b*x+8*b^2))/(105*a^3)*sqrt((a+b*x)^3)
--R
--R
--R +----------------------------+
---R 2 2 2 | 3 3 2 2 2 3
---R (30a x - 24a b x + 16b )\|a x + 3a b x + 3a b x + b
+--R 2 2 2 | 3 3 2 2 2 3
+--R (30a x - 24a b x + 16b )\|b x + 3a b x + 3a b x + a
--R (2) --------------------------------------------------------
---R 2
+--R 3
--R 105a
--R Type: Expression
Integer
--E
---S 59 14:91 Axiom cannot simplify this expression
+--S 55 14:91 Axiom cannot simplify this expression. Schaums typo?
cc:=aa-bb
---R
--R
--R (3)
---R +----------------------------+
---R 3 2 2 2 | 3 3 2 2 2 3
---R (- 30a x + 24a b x - 16a b )\|a x + 3a b x + 3a b x + b
+--R +----------------------------+
+--R 2 2 2 | 3 3 2 2 2 3
+--R (- 30a x + 24a b x - 16b )\|b x + 3a b x + 3a b x + a
--R +
--R 3 3 2 2 2 3 +-------+
--R (30a x + 6a b x - 8a b x + 16b )\|a x + b
@@ -848,6 +800,20 @@ cc:=aa-bb
--R 105a
--R Type: Expression
Integer
--E
+
+@
+Notice that if we factor the numerator of 'aa' we get an expression that
+differs from schaums on by the order of the variables in the square root.
+(We can square the term (a*x+b) and drag it under the square root to get
+the cubic term). It appears that Schaums has a typo.
+<<*>>=
+--S 56
+factor numer aa
+--R
+--R 2 2 2 +-------+
+--R (4) 2(a x + b)(15a x - 12a b x + 8b )\|a x + b
+--RType: Factored SparseMultivariatePolynomial(Integer,Kernel Expression
Integer)
+--E
@
\section{\cite{1}:14.92~~~~~$\displaystyle
@@ -858,7 +824,7 @@ $$
<<*>>=
)clear all
---S 60
+--S 57
aa:=integrate(sqrt(a*x+b)/x,x)
--R
--R
@@ -875,7 +841,7 @@ aa:=integrate(sqrt(a*x+b)/x,x)
--R Type: Union(List Expression
Integer,...)
--E
---S 61
+--S 58
dd:=integrate(1/(x*sqrt(a*x+b)),x)
--R
--R
@@ -889,7 +855,7 @@ dd:=integrate(1/(x*sqrt(a*x+b)),x)
--R Type: Union(List Expression
Integer,...)
--E
---S 62
+--S 59
bb1:=2*sqrt(a*x+b)+b*dd.1
--R
--R
@@ -903,7 +869,7 @@ bb1:=2*sqrt(a*x+b)+b*dd.1
--R Type: Expression
Integer
--E
---S 63
+--S 60
bb2:=2*sqrt(a*x+b)+b*dd.2
--R
--R
@@ -917,7 +883,7 @@ bb2:=2*sqrt(a*x+b)+b*dd.2
--R Type: Expression
Integer
--E
---S 64
+--S 61
cc11:=bb1-aa.1
--R
--R
@@ -932,7 +898,7 @@ cc11:=bb1-aa.1
--R Type: Expression
Integer
--E
---S 65
+--S 62
cc12:=bb1-aa.2
--R
--R
@@ -947,7 +913,7 @@ cc12:=bb1-aa.2
--R Type: Expression
Integer
--E
---S 66
+--S 63
cc21:=bb2-aa.1
--R
--R
@@ -962,7 +928,7 @@ cc21:=bb2-aa.1
--R Type: Expression
Integer
--E
---S 67 14:92 Axiom cannot simplify this expression
+--S 64 14:92 Axiom cannot simplify this expression
cc22:=bb2-aa.2
--R
--R
@@ -986,7 +952,7 @@ $$
<<*>>=
)clear all
---S 68
+--S 65
aa:=integrate(sqrt(a*x+b)/x^2,x)
--R
--R
@@ -1008,7 +974,7 @@ aa:=integrate(sqrt(a*x+b)/x^2,x)
--R Type: Union(List Expression
Integer,...)
--E
---S 69
+--S 66
dd:=integrate(1/(x*sqrt(a*x+b)),x)
--R
--R
@@ -1022,7 +988,7 @@ dd:=integrate(1/(x*sqrt(a*x+b)),x)
--R Type: Union(List Expression
Integer,...)
--E
---S 70
+--S 67
bb1:=-sqrt(a*x+b)/x+a/2*dd.1
--R
--R
@@ -1036,7 +1002,7 @@ bb1:=-sqrt(a*x+b)/x+a/2*dd.1
--R Type: Expression
Integer
--E
---S 71
+--S 68
bb2:=-sqrt(a*x+b)/x+a/2*dd.2
--R
--R
@@ -1050,7 +1016,7 @@ bb2:=-sqrt(a*x+b)/x+a/2*dd.2
--R Type: Expression
Integer
--E
---S 72
+--S 69
cc11:=bb1-aa.1
--R
--R
@@ -1058,7 +1024,7 @@ cc11:=bb1-aa.1
--R Type: Expression
Integer
--E
---S 73
+--S 70
cc21:=bb-aa.1
--R
--R
@@ -1073,7 +1039,7 @@ cc21:=bb-aa.1
--R Type: Expression
Integer
--E
---S 74
+--S 71
cc12:=bb1-aa.2
--R
--R
@@ -1088,7 +1054,7 @@ cc12:=bb1-aa.2
--R Type: Expression
Integer
--E
---S 75 14:93 Schaums and Axiom agree
+--S 72 14:93 Schaums and Axiom agree
cc22:=bb2-aa.2
--R
--R
@@ -1106,7 +1072,7 @@ $$
<<*>>=
)clear all
---S 76 14:94 Axiom cannot do this integral
+--S 73 14:94 Axiom cannot do this integral
aa:=integrate(x^m/sqrt(a*x+b),x)
--R
--R
@@ -1128,7 +1094,7 @@ $$
<<*>>=
)clear all
---S 77 14:95 Axiom cannot do this integral
+--S 74 14:95 Axiom cannot do this integral
aa:=integrate(1/(x^m*sqrt(a*x+b)),x)
--R
--R
@@ -1150,7 +1116,7 @@ $$
<<*>>=
)clear all
---S 78 14:96 Axiom cannot do this integral
+--S 75 14:96 Axiom cannot do this integral
aa:=integrate(x^m*sqrt(a*x+b),x)
--R
--R
@@ -1171,7 +1137,7 @@ $$
<<*>>=
)clear all
---S 79 14:97 Axiom cannot do this integral
+--S 76 14:97 Axiom cannot do this integral
aa:=integrate(sqrt(a*x+b)/x^m,x)
--R
--R
@@ -1194,7 +1160,7 @@ Note: 14.98 is the same as 14.97
<<*>>=
)clear all
---S 80 14:98 Axiom cannot do this integral
+--S 77 14:98 Axiom cannot do this integral
aa:=integrate(sqrt(a*x+b)/x^m,x)
--R
--R
@@ -1215,7 +1181,7 @@ $$
<<*>>=
)clear all
---S 81
+--S 78
aa:=integrate((a*x+b)^(m/2),x)
--R
--R
@@ -1228,7 +1194,7 @@ aa:=integrate((a*x+b)^(m/2),x)
--R Type: Union(Expression
Integer,...)
--E
---S 82
+--S 79
bb:=(2*(a*x+b)^((m+2)/2))/(a*(m+2))
--R
--R
@@ -1241,7 +1207,7 @@ bb:=(2*(a*x+b)^((m+2)/2))/(a*(m+2))
--R Type: Expression
Integer
--E
---S 83 14:99 Axiom cannot simplify this expression
+--S 80
cc:=aa-bb
--R
--R
@@ -1253,6 +1219,33 @@ cc:=aa-bb
--R a m + 2a
--R Type: Expression
Integer
--E
+
+--S 81
+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 82
+dd:=explog cc
+--R
+--R m + 2 m
+--R ----- -
+--R 2 2
+--R - 2(a x + b) + (2a x + 2b)(a x + b)
+--R (5) -----------------------------------------
+--R a m + 2a
+--R Type: Expression
Integer
+--E
+
+--S 83 14:99 Schaums and Axiom agree
+ee:=complexNormalize dd
+--R
+--R (6) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.100~~~~~$\displaystyle
@@ -1292,7 +1285,7 @@
bb:=(2*(a*x+b)^((m+4)/2))/(a^2*(m+4))-(2*b*(a*x+b)^((m+2)/2))/(a^2*(m+2))
--R Type: Expression
Integer
--E
---S 86 14:100 Axiom cannot simplify this expression
+--S 86
cc:=aa-bb
--R
--R
@@ -1311,6 +1304,40 @@ cc:=aa-bb
--R a m + 6a m + 8a
--R Type: Expression
Integer
--E
+
+--S 87
+explog:=rule(%e^(n*log(x)) == x^n)
+--R
+--R n log(x) n
+--R (4) %e == x
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 88
+dd:=explog cc
+--R
+--R (5)
+--R m + 4 m + 2
+--R ----- -----
+--R 2 2
+--R (- 2m - 4)(a x + b) + (2b m + 8b)(a x + b)
+--R +
+--R m
+--R -
+--R 2 2 2 2 2
+--R ((2a m + 4a )x + 2a b m x - 4b )(a x + b)
+--R /
+--R 2 2 2 2
+--R a m + 6a m + 8a
+--R Type: Expression
Integer
+--E
+
+--S 89 14:100 Schaums and Axiom agree
+ee:=complexNormalize dd
+--R
+--R (6) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.101~~~~~$\displaystyle
@@ -1323,7 +1350,7 @@ $$
<<*>>=
)clear all
---S 87
+--S 90
aa:=integrate(x^2*(a*x+b)^(m/2),x)
--R
--R
@@ -1341,7 +1368,7 @@ aa:=integrate(x^2*(a*x+b)^(m/2),x)
--R Type: Union(Expression
Integer,...)
--E
---S 88
+--S 91
bb:=(2*(a*x+b)^((m+6)/2))/(a^3*(m+6))-_
(4*b*(a*x+b)^((m+4)/2))/(a^3*(m+4))+_
(2*b^2*(a*x+b)^((m+2)/2))/(a^3*(m+2))
@@ -1363,7 +1390,7 @@ bb:=(2*(a*x+b)^((m+6)/2))/(a^3*(m+6))-_
--R Type: Expression
Integer
--E
---S 89 14:101 Axiom cannot simplify this expression
+--S 92
cc:=aa-bb
--R
--R
@@ -1390,6 +1417,48 @@ cc:=aa-bb
--R a m + 12a m + 44a m + 48a
--R Type: Expression
Integer
--E
+
+--S 93
+explog:=rule(%e^(n*log(x)) == x^n)
+--R
+--R n log(x) n
+--R (4) %e == x
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 94
+dd:=explog cc
+--R
+--R (5)
+--R m + 6 m +
4
+--R -----
-----
+--R 2 2 2 2
+--R (- 2m - 12m - 16)(a x + b) + (4b m + 32b m + 48b)(a x + b)
+--R +
+--R m + 2
+--R -----
+--R 2 2 2 2 2
+--R (- 2b m - 20b m - 48b )(a x + b)
+--R +
+--R 3 2 3 3 3 2 2 2 2 2 3
+--R ((2a m + 12a m + 16a )x + (2a b m + 4a b m)x - 8a b m x + 16b )
+--R *
+--R m
+--R -
+--R 2
+--R (a x + b)
+--R /
+--R 3 3 3 2 3 3
+--R a m + 12a m + 44a m + 48a
+--R Type: Expression
Integer
+--E
+
+--S 95 14:101 Schaums and Axiom agree
+ee:=complexNormalize dd
+--R
+--R (6) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.102~~~~~$\displaystyle
@@ -1401,7 +1470,7 @@ $$
<<*>>=
)clear all
---S 90 14:102 Axiom cannot do this integral
+--S 96 14:102 Axiom cannot do this integral
aa:=integrate((a*x+b)^(m/2)/x,x)
--R
--R
@@ -1423,7 +1492,7 @@ $$
<<*>>=
)clear all
---S 91 14:103 Axiom cannot do this integral
+--S 97 14:103 Axiom cannot do this integral
aa:=integrate((a*x+b)^(m/2)/x^2,x)
--R
--R
@@ -1446,7 +1515,7 @@ $$
<<*>>=
)clear all
---S 92 14:104 Axiom cannot do this integral
+--S 98 14:104 Axiom cannot do this integral
aa:=integrate(1/(x*(a*x+b)^(m/2)),x)
--R
--R
diff --git a/src/input/schaum6.input.pamphlet b/src/input/schaum6.input.pamphlet
index 9b6f4f0..854e780 100644
--- a/src/input/schaum6.input.pamphlet
+++ b/src/input/schaum6.input.pamphlet
@@ -792,7 +792,7 @@ explog:=rule(%e^(n*log(x)) == x^n)
--R Type: RewriteRule(Integer,Integer,Expression
Integer)
--E
---S 64 14:140 Schaums and Axiom cannot simplify this expression
+--S 64
dd:=explog cc
--R
--R 2 2 n 2 2 2 2 n - 1
@@ -802,6 +802,13 @@ dd:=explog cc
--R (2n - 2)(x + a ) (x + a )
--R Type: Expression
Integer
--E
+
+--S 65 14:140 Schaums and Axiom agree
+ee:=complexNormalize dd
+--R
+--R (6) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.141~~~~~$\displaystyle\int{\frac{dx}{x(x^2+a^2)^n}}$}
@@ -812,7 +819,7 @@ $$
<<*>>=
)clear all
---S 65 14:141 Axiom cannot do this integral
+--S 66 14:141 Axiom cannot do this integral
aa:=integrate(1/(x*(x^2+a^2)^n),x)
--R
--R
@@ -833,7 +840,7 @@ $$
<<*>>=
)clear all
---S 66 14:142 Axiom cannot do this integral
+--S 67 14:142 Axiom cannot do this integral
aa:=integrate(x^m/((x^2+a^2)^n),x)
--R
--R
@@ -854,7 +861,7 @@ $$
<<*>>=
)clear all
---S 67 14:143 Axiom cannot do this integral
+--S 68 14:143 Axiom cannot do this integral
aa:=integrate(1/(x^m*(x^2+a^2)^n),x)
--R
--R
diff --git a/src/input/schaum7.input.pamphlet b/src/input/schaum7.input.pamphlet
index ce37f19..fcdd71c 100644
--- a/src/input/schaum7.input.pamphlet
+++ b/src/input/schaum7.input.pamphlet
@@ -915,7 +915,7 @@ explog:=rule(%e^(n*log(x)) == x^n)
--R Type: RewriteRule(Integer,Integer,Expression
Integer)
--E
---S 76 14:159 Axiom cannot simplify this expression
+--S 76
dd:=explog cc
--R
--R 2 2 n 2 2 2 2 n - 1
@@ -926,6 +926,12 @@ dd:=explog cc
--R Type: Expression
Integer
--E
+--S 77 14:159 Schaums and Axiom agree
+ee:=complexNormalize dd
+--R
+--R (6) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.160~~~~~$\displaystyle\int{\frac{dx}{x(x^2-a^2)^n}}$}
@@ -936,7 +942,7 @@ $$
<<*>>=
)clear all
---S 77 14:160 Axiom cannot compute this integral
+--S 78 14:160 Axiom cannot compute this integral
aa:=integrate(1/(x*(x^2-a^2)^n),x)
--R
--R
@@ -957,7 +963,7 @@ $$
<<*>>=
)clear all
---S 78 14:161 Axiom cannot compute this integral
+--S 79 14:161 Axiom cannot compute this integral
aa:=integrate(x^m/((x^2-a^2)^n),x)
--R
--R
@@ -978,7 +984,7 @@ $$
<<*>>=
)clear all
---S 79 14:162 Axiom cannot compute this integral
+--S 80 14:162 Axiom cannot compute this integral
aa:=integrate(1/(x^m*(x^2-a^2)^n),x)
--R
--R
diff --git a/src/input/schaum8.input.pamphlet b/src/input/schaum8.input.pamphlet
index 778d135..10d9ef5 100644
--- a/src/input/schaum8.input.pamphlet
+++ b/src/input/schaum8.input.pamphlet
@@ -1103,7 +1103,7 @@ explog:=rule(%e^(n*log(x)) == x^n)
--R Type: RewriteRule(Integer,Integer,Expression
Integer)
--E
---S 95 14:178 Axiom cannot simplify this expression
+--S 95
dd:=explog cc
--R
--R 2 2 n 2 2 2 2 n - 1
@@ -1114,6 +1114,13 @@ dd:=explog cc
--R Type: Expression
Integer
--E
+--S 96 14:178 Schaums and Axiom agree
+ee:=complexNormalize dd
+--R
+--R (6) 0
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.179~~~~~$\displaystyle\int{\frac{dx}{x(a^2-x^2)^n}}$}
@@ -1124,7 +1131,7 @@ $$
<<*>>=
)clear all
---S 96 14:179 Axiom cannot integrate this expression
+--S 97 14:179 Axiom cannot integrate this expression
aa:=integrate(1/(x*(a^2-x^2)^n),x)
--R
--R
@@ -1145,7 +1152,7 @@ $$
<<*>>=
)clear all
---S 97 14:180 Axiom cannot integrate this expression
+--S 98 14:180 Axiom cannot integrate this expression
aa:=integrate(x^m/((a^2-x^2)^n),x)
--R
--R
@@ -1166,7 +1173,7 @@ $$
<<*>>=
)clear all
---S 98 14:181 Axiom cannot integrate this expression
+--S 99 14:181 Axiom cannot integrate this expression
aa:=integrate(1/(x^m*(a^2-x^2)^n),x)
--R
--R
diff --git a/src/input/schaum9.input.pamphlet b/src/input/schaum9.input.pamphlet
index fceb094..d632de7 100644
--- a/src/input/schaum9.input.pamphlet
+++ b/src/input/schaum9.input.pamphlet
@@ -35,7 +35,7 @@ bb:=log(x+sqrt(x^2+a^2))
--R Type: Expression
Integer
--E
---S 3 14:182 Axiom cannot simplify this expression
+--S 3
cc:=aa-bb
--R
--R +-------+ +-------+
@@ -43,6 +43,15 @@ cc:=aa-bb
--R (3) - log(\|x + a + x) - log(\|x + a - x)
--R Type: Expression
Integer
--E
+
+--S 4 14:182 Schaums and Axiom differ by a constant
+dd:=complexNormalize cc
+--R
+--R 2
+--R (4) - log(a )
+--R Type: Expression
Integer
+--E
+
@
This is equal to $-\log(a^2)$ but Axiom cannot prove it.
@@ -51,7 +60,7 @@ $$\int{\frac{x}{\sqrt{x^2+a^2}}}=\sqrt{x^2+a^2}$$
<<*>>=
)clear all
---S 4
+--S 5
aa:=integrate(x/(sqrt(x^2+a^2)),x)
--R
--R
@@ -65,7 +74,7 @@ aa:=integrate(x/(sqrt(x^2+a^2)),x)
--R Type: Union(Expression
Integer,...)
--E
---S 5
+--S 6
bb:=sqrt(x^2+a^2)
--R
--R +-------+
@@ -74,7 +83,7 @@ bb:=sqrt(x^2+a^2)
--R Type: Expression
Integer
--E
---S 6 14:183 Schaums and Axiom agree
+--S 7 14:183 Schaums and Axiom agree
cc:=aa-bb
--R
--R (3) 0
@@ -90,7 +99,7 @@ $$
<<*>>=
)clear all
---S 7
+--S 8
aa:=integrate(x^2/sqrt(x^2+a^2),x)
--R
--R
@@ -109,7 +118,7 @@ aa:=integrate(x^2/sqrt(x^2+a^2),x)
--R Type: Union(Expression
Integer,...)
--E
---S 8
+--S 9
bb:=(x*sqrt(x^2+a^2))/2-a^2/2*log(x+sqrt(x^2+a^2))
--R
--R +-------+ +-------+
@@ -120,7 +129,7 @@ bb:=(x*sqrt(x^2+a^2))/2-a^2/2*log(x+sqrt(x^2+a^2))
--R Type: Expression
Integer
--E
---S 9
+--S 10
cc:=aa-bb
--R
--R +-------+ +-------+
@@ -131,14 +140,14 @@ cc:=aa-bb
--R Type: Expression
Integer
--E
---S 10
+--S 11
logmul1:=rule(c*log(a)+c*log(b) == c*log(a*b))
--R
--I (4) c log(b) + c log(a) + %K == c log(a b) + %K
--R Type: RewriteRule(Integer,Integer,Expression
Integer)
--E
---S 11 14:184 Schaums and Axiom differ by a constant
+--S 12 14:184 Schaums and Axiom differ by a constant
dd:=logmul1 cc
--R
--R 2 2
@@ -158,7 +167,7 @@ $$
<<*>>=
)clear all
---S 12
+--S 13
aa:=integrate(x^3/sqrt(x^2+a^2),x)
--R
--R
@@ -172,7 +181,7 @@ aa:=integrate(x^3/sqrt(x^2+a^2),x)
--R Type: Union(Expression
Integer,...)
--E
---S 13
+--S 14
bb:=(x^2+a^2)^(3/2)/3-a^2*sqrt(x^2+a^2)
--R
--R +-------+
@@ -183,7 +192,7 @@ bb:=(x^2+a^2)^(3/2)/3-a^2*sqrt(x^2+a^2)
--R Type: Expression
Integer
--E
---S 14 14:185 Schaums and Axiom agree
+--S 15 14:185 Schaums and Axiom agree
cc:=aa-bb
--R
--R (3) 0
@@ -198,7 +207,7 @@ $$
<<*>>=
)clear all
---S 15
+--S 16
aa:=integrate(1/(x*sqrt(x^2+a^2)),x)
--R
--R
@@ -210,7 +219,7 @@ aa:=integrate(1/(x*sqrt(x^2+a^2)),x)
--R Type: Union(Expression
Integer,...)
--E
---S 16
+--S 17
bb:=-1/a*log((a+sqrt(x^2+a^2))/x)
--R
--R +-------+
@@ -223,7 +232,7 @@ bb:=-1/a*log((a+sqrt(x^2+a^2))/x)
--R Type: Expression
Integer
--E
---S 17
+--S 18
cc:=aa-bb
--R
--R (3)
@@ -237,19 +246,10 @@ cc:=aa-bb
--R Type: Expression
Integer
--E
---S 18
-divlog:=rule(log(a/b) == log(a) - log(b))
---R
---R a
---R (4) log(-) == - log(b) + log(a)
---R b
---R Type: RewriteRule(Integer,Integer,Expression
Integer)
---E
-
---S 19 14:186 Axiom cannot simplify this expression
-dd:=divlog cc
+--S 19
+dd:=expandLog cc
--R
---R (5)
+--R (4)
--R +-------+ +-------+ +-------+
--R | 2 2 | 2 2 | 2 2
--R log(\|x + a + a) - log(\|x + a - x + a) + log(\|x + a - x -
a)
@@ -260,6 +260,15 @@ dd:=divlog cc
--R Type: Expression
Integer
--E
+--S 20 14:186 Schaums and Axiom differ by a constant
+ee:=complexNormalize dd
+--R
+--R log(- 1)
+--R (5) - --------
+--R a
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.187~~~~~$\displaystyle
@@ -270,7 +279,7 @@ $$
<<*>>=
)clear all
---S 20
+--S 21
aa:=integrate(1/(x^2*sqrt(x^2+a^2)),x)
--R
--R
@@ -282,7 +291,7 @@ aa:=integrate(1/(x^2*sqrt(x^2+a^2)),x)
--R Type: Union(Expression
Integer,...)
--E
---S 21
+--S 22
bb:=-sqrt(x^2+a^2)/(a^2*x)
--R
--R +-------+
@@ -294,7 +303,7 @@ bb:=-sqrt(x^2+a^2)/(a^2*x)
--R Type: Expression
Integer
--E
---S 22 14:187 Schaums and Axiom differ by a constant
+--S 23 14:187 Schaums and Axiom differ by a constant
cc:=aa-bb
--R
--R 1
@@ -314,7 +323,7 @@ $$
<<*>>=
)clear all
---S 23
+--S 24
aa:=integrate(1/(x^3*sqrt(x^2+a^2)),x)
--R
--R
@@ -337,7 +346,7 @@ aa:=integrate(1/(x^3*sqrt(x^2+a^2)),x)
--R Type: Union(Expression
Integer,...)
--E
---S 24
+--S 25
bb:=-sqrt(x^2+a^2)/(2*a^2*x^2)+1/(2*a^3)*log((a+sqrt(x^2+a^2))/x)
--R
--R +-------+
@@ -351,7 +360,7 @@
bb:=-sqrt(x^2+a^2)/(2*a^2*x^2)+1/(2*a^3)*log((a+sqrt(x^2+a^2))/x)
--R Type: Expression
Integer
--E
---S 25
+--S 26
cc:=aa-bb
--R
--R (3)
@@ -366,19 +375,10 @@ cc:=aa-bb
--R Type: Expression
Integer
--E
---S 26
-divlog:=rule(log(a/b) == log(a) - log(b))
+--S 27
+dd:=expandLog cc
--R
---R a
---R (4) log(-) == - log(b) + log(a)
---R b
---R Type: RewriteRule(Integer,Integer,Expression
Integer)
---E
-
---S 27 14:188 Axiom cannot simplify this expression
-dd:=divlog cc
---R
---R (5)
+--R (4)
--R +-------+ +-------+ +-------+
--R | 2 2 | 2 2 | 2 2
--R - log(\|x + a + a) + log(\|x + a - x + a) - log(\|x + a - x
- a)
@@ -389,6 +389,17 @@ dd:=divlog cc
--R 2a
--R Type: Expression
Integer
--E
+
+--S 28 14:188 Schaums and Axiom differ by a constant
+ee:=complexNormalize dd
+
+ log(- 1)
+ (5) --------
+ 3
+ 2a
+ Type: Expression Integer
+--E
+
@
\section{\cite{1}:14.189~~~~~$\displaystyle\int{\sqrt{x^2+a^2}}~dx$}
@@ -398,7 +409,7 @@ $$
<<*>>=
)clear all
---S 28
+--S 29
aa:=integrate(sqrt(x^2+a^2),x)
--R
--R
@@ -417,7 +428,7 @@ aa:=integrate(sqrt(x^2+a^2),x)
--R Type: Union(Expression
Integer,...)
--E
---S 29
+--S 30
bb:=(x*sqrt(x^2+a^2))/2+a^2/2*log(x+sqrt(x^2+a^2))
--R
--R +-------+ +-------+
@@ -428,7 +439,7 @@ bb:=(x*sqrt(x^2+a^2))/2+a^2/2*log(x+sqrt(x^2+a^2))
--R Type: Expression
Integer
--E
---S 30 14:189 Axiom cannot simplify this expression
+--S 31
cc:=aa-bb
--R
--R +-------+ +-------+
@@ -438,6 +449,17 @@ cc:=aa-bb
--R 2
--R Type: Expression
Integer
--E
+
+--S 32 14:189 Schaums and Axiom differ by a constant
+dd:=complexNormalize cc
+--R
+--R 2 2
+--R a log(a )
+--R (4) - ---------
+--R 2
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.190~~~~~$\displaystyle\int{x\sqrt{x^2+a^2}}~dx$}
@@ -447,7 +469,7 @@ $$
<<*>>=
)clear all
---S 31
+--S 33
aa:=integrate(x*sqrt(x^2+a^2),x)
--R
--R
@@ -461,7 +483,7 @@ aa:=integrate(x*sqrt(x^2+a^2),x)
--R Type: Union(Expression
Integer,...)
--E
---S 32
+--S 34
bb:=(x^2+a^2)^(3/2)/3
--R
--R +-------+
@@ -472,7 +494,7 @@ bb:=(x^2+a^2)^(3/2)/3
--R Type: Expression
Integer
--E
---S 33 14:190 Schaums and Axiom agree
+--S 35 14:190 Schaums and Axiom agree
cc:=aa-bb
--R
--R (3) 0
@@ -489,7 +511,7 @@ $$
<<*>>=
)clear all
---S 34
+--S 36
aa:=integrate(x^2*sqrt(x^2+a^2),x)
--R
--R
@@ -508,7 +530,7 @@ aa:=integrate(x^2*sqrt(x^2+a^2),x)
--R Type: Union(Expression
Integer,...)
--E
---S 35
+--S 37
bb:=(x*(x^2+a^2)^(3/2))/4-(a^2*x*sqrt(x^2+a^2))/8-a^4/8*log(x+sqrt(x^2+a^2))
--R
--R +-------+ +-------+
@@ -519,7 +541,7 @@
bb:=(x*(x^2+a^2)^(3/2))/4-(a^2*x*sqrt(x^2+a^2))/8-a^4/8*log(x+sqrt(x^2+a^2))
--R Type: Expression
Integer
--E
---S 36 14:191 Axiom cannot simplify this expression
+--S 38
cc:=aa-bb
--R
--R +-------+ +-------+
@@ -529,6 +551,16 @@ cc:=aa-bb
--R 8
--R Type: Expression
Integer
--E
+
+--S 39 14:191 Schaums and Axiom differ by a constant
+dd:=complexNormalize cc
+--R
+--R 4 2
+--R a log(a )
+--R (4) ---------
+--R 8
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.192~~~~~$\displaystyle
@@ -539,7 +571,7 @@ $$
<<*>>=
)clear all
---S 37
+--S 40
aa:=integrate(x^3*sqrt(x^2+a^2),x)
--R
--R
@@ -557,7 +589,7 @@ aa:=integrate(x^3*sqrt(x^2+a^2),x)
--R Type: Union(Expression
Integer,...)
--E
---S 38
+--S 41
bb:=(x^2+a^2)^(5/2)/5-(a^2*(x^2+a^2)^(3/2))/3
--R
--R +-------+
@@ -568,7 +600,7 @@ bb:=(x^2+a^2)^(5/2)/5-(a^2*(x^2+a^2)^(3/2))/3
--R Type: Expression
Integer
--E
---S 39 14:192 Schaums and Axiom agree
+--S 42 14:192 Schaums and Axiom agree
cc:=aa-bb
--R
--R (3) 0
@@ -584,7 +616,7 @@ $$
<<*>>=
)clear all
---S 40
+--S 43
aa:=integrate(sqrt(x^2+a^2)/x,x)
--R
--R
@@ -603,7 +635,7 @@ aa:=integrate(sqrt(x^2+a^2)/x,x)
--R Type: Union(Expression
Integer,...)
--E
---S 41
+--S 44
bb:=sqrt(x^2+a^2)-a*log((a+sqrt(x^2+a^2))/x)
--R
--R +-------+
@@ -614,7 +646,7 @@ bb:=sqrt(x^2+a^2)-a*log((a+sqrt(x^2+a^2))/x)
--R Type: Expression
Integer
--E
---S 42 14:193 Axiom cannot simplify this expression
+--S 45
cc:=aa-bb
--R
--R (3)
@@ -629,6 +661,27 @@ cc:=aa-bb
--R x
--R Type: Expression
Integer
--E
+
+--S 46
+dd:=expandLog cc
+--R
+--R (4)
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R a log(\|x + a + a) - a log(\|x + a - x + a)
+--R +
+--R +-------+
+--R | 2 2
+--R a log(\|x + a - x - a) - a log(x)
+--R Type: Expression
Integer
+--E
+
+--S 47 14:193 Schaums and Axiom differ by a constant
+ee:=complexNormalize dd
+--R
+--R (5) - a log(- 1)
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.194~~~~~$\displaystyle
@@ -639,7 +692,7 @@ $$
<<*>>=
)clear all
---S 43
+--S 48
aa:=integrate(sqrt(x^2+a^2)/x^2,x)
--R
--R
@@ -653,7 +706,7 @@ aa:=integrate(sqrt(x^2+a^2)/x^2,x)
--R Type: Union(Expression
Integer,...)
--E
---S 44
+--S 49
bb:=-sqrt(x^2+a^2)/x+log(x+sqrt(x^2+a^2))
--R
--R +-------+ +-------+
@@ -664,7 +717,7 @@ bb:=-sqrt(x^2+a^2)/x+log(x+sqrt(x^2+a^2))
--R Type: Expression
Integer
--E
---S 45 14:194 Axiom cannot simplify this expression
+--S 50
cc:=aa-bb
--R
--R +-------+ +-------+
@@ -672,6 +725,15 @@ cc:=aa-bb
--R (3) - log(\|x + a + x) - log(\|x + a - x) - 1
--R Type: Expression
Integer
--E
+
+--S 51 14:194 Schaums and Axiom differ by a constant
+dd:=complexNormalize cc
+--R
+--R 2
+--R (4) - log(a ) - 1
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.195~~~~~$\displaystyle
@@ -683,7 +745,7 @@ $$
<<*>>=
)clear all
---S 46
+--S 52
aa:=integrate(sqrt(x^2+a^2)/x^3,x)
--R
--R
@@ -706,7 +768,7 @@ aa:=integrate(sqrt(x^2+a^2)/x^3,x)
--R Type: Union(Expression
Integer,...)
--E
---S 47
+--S 53
bb:=-sqrt(x^2+a^2)/(2*x^2)-1/(2*a)*log((a+sqrt(x^2+a^2))/x)
--R
--R +-------+
@@ -720,7 +782,7 @@ bb:=-sqrt(x^2+a^2)/(2*x^2)-1/(2*a)*log((a+sqrt(x^2+a^2))/x)
--R Type: Expression
Integer
--E
---S 48
+--S 54
cc:=aa-bb
--R
--R (3)
@@ -734,19 +796,10 @@ cc:=aa-bb
--R Type: Expression
Integer
--E
---S 49
-divlog:=rule(log(a/b) == log(a) - log(b))
---R
---R a
---R (4) log(-) == - log(b) + log(a)
---R b
---R Type: RewriteRule(Integer,Integer,Expression
Integer)
---E
-
---S 50 14:195 Axiom cannot simplify this expression
-dd:=divlog cc
+--S 55
+dd:=expandLog cc
--R
---R (5)
+--R (4)
--R +-------+ +-------+ +-------+
--R | 2 2 | 2 2 | 2 2
--R log(\|x + a + a) - log(\|x + a - x + a) + log(\|x + a - x -
a)
@@ -756,6 +809,15 @@ dd:=divlog cc
--R 2a
--R Type: Expression
Integer
--E
+
+--S 56 14:195 Schaums and Axiom differ by a constant
+ee:=complexNormalize dd
+--R
+--R log(- 1)
+--R (5) - --------
+--R 2a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.196~~~~~$\displaystyle\int{\frac{dx}{(x^2+a^2)^{3/2}}}$}
@@ -765,7 +827,7 @@ $$
<<*>>=
)clear all
---S 51
+--S 57
aa:=integrate(1/(x^2+a^2)^(3/2),x)
--R
--R
@@ -777,7 +839,7 @@ aa:=integrate(1/(x^2+a^2)^(3/2),x)
--R Type: Union(Expression
Integer,...)
--E
---S 52
+--S 58
bb:=x/(a^2*sqrt(x^2+a^2))
--R
--R x
@@ -788,7 +850,7 @@ bb:=x/(a^2*sqrt(x^2+a^2))
--R Type: Expression
Integer
--E
---S 53 14:196 Schaums and Axiom differ by a constant
+--S 59 14:196 Schaums and Axiom differ by a constant
cc:=aa-bb
--R
--R 1
@@ -807,7 +869,7 @@ $$
<<*>>=
)clear all
---S 54
+--S 60
aa:=integrate(x/(x^2+a^2)^(3/2),x)
--R
--R
@@ -821,7 +883,7 @@ aa:=integrate(x/(x^2+a^2)^(3/2),x)
--R Type: Union(Expression
Integer,...)
--E
---S 55
+--S 61
bb:=-1/sqrt(x^2+a^2)
--R
--R 1
@@ -832,7 +894,7 @@ bb:=-1/sqrt(x^2+a^2)
--R Type: Expression
Integer
--E
---S 56 14:197 Schaums and Axiom agree
+--S 62 14:197 Schaums and Axiom agree
cc:=aa-bb
--R
--R (3) 0
@@ -849,7 +911,7 @@ $$
<<*>>=
)clear all
---S 57
+--S 63
aa:=integrate(x^2/(x^2+a^2)^(3/2),x)
--R
--R
@@ -863,7 +925,7 @@ aa:=integrate(x^2/(x^2+a^2)^(3/2),x)
--R Type: Union(Expression
Integer,...)
--E
---S 58
+--S 64
bb:=-x/sqrt(x^2+a^2)+log(x+sqrt(x^2+a^2))
--R
--R +-------+ +-------+
@@ -876,7 +938,7 @@ bb:=-x/sqrt(x^2+a^2)+log(x+sqrt(x^2+a^2))
--R Type: Expression
Integer
--E
---S 59 14:198 Axiom cannot simplify this expression
+--S 65
cc:=aa-bb
--R
--R +-------+ +-------+
@@ -884,6 +946,14 @@ cc:=aa-bb
--R (3) - log(\|x + a + x) - log(\|x + a - x) - 1
--R Type: Expression
Integer
--E
+
+--S 66 14:198 Schaums and Axiom differ by a constant
+dd:=complexNormalize cc
+--R
+--R 2
+--R (4) - log(a ) - 1
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.199~~~~~$\displaystyle
@@ -894,7 +964,7 @@ $$
<<*>>=
)clear all
---S 60
+--S 67
aa:=integrate(x^3/(x^2+a^2)^(3/2),x)
--R
--R
@@ -908,7 +978,7 @@ aa:=integrate(x^3/(x^2+a^2)^(3/2),x)
--R Type: Union(Expression
Integer,...)
--E
---S 61
+--S 68
bb:=sqrt(x^2+a^2)+a^2/sqrt(x^2+a^2)
--R
--R 2 2
@@ -920,7 +990,7 @@ bb:=sqrt(x^2+a^2)+a^2/sqrt(x^2+a^2)
--R Type: Expression
Integer
--E
---S 62 14:199 Schaums and Axiom agree
+--S 69 14:199 Schaums and Axiom agree
cc:=aa-bb
--R
--R (3) 0
@@ -937,7 +1007,7 @@ $$
<<*>>=
)clear all
---S 63
+--S 70
aa:=integrate(1/(x*(x^2+a^2)^(3/2)),x)
--R
--R
@@ -956,7 +1026,7 @@ aa:=integrate(1/(x*(x^2+a^2)^(3/2)),x)
--R Type: Union(Expression
Integer,...)
--E
---S 64
+--S 71
bb:=1/(a^2*sqrt(x^2+a^2))-1/a^3*log((a+sqrt(x^2+a^2))/x)
--R
--R +-------+
@@ -971,7 +1041,7 @@ bb:=1/(a^2*sqrt(x^2+a^2))-1/a^3*log((a+sqrt(x^2+a^2))/x)
--R Type: Expression
Integer
--E
---S 65
+--S 72
cc:=aa-bb
--R
--R (3)
@@ -986,19 +1056,10 @@ cc:=aa-bb
--R Type: Expression
Integer
--E
---S 66
-divlog:=rule(log(a/b) == log(a) - log(b))
---R
---R a
---R (4) log(-) == - log(b) + log(a)
---R b
---R Type: RewriteRule(Integer,Integer,Expression
Integer)
---E
-
---S 67 14:200 Axiom cannot simplify this expression
-dd:=divlog cc
+--S 73
+dd:=expandLog cc
--R
---R (5)
+--R (4)
--R +-------+ +-------+ +-------+
--R | 2 2 | 2 2 | 2 2
--R log(\|x + a + a) - log(\|x + a - x + a) + log(\|x + a - x -
a)
@@ -1009,6 +1070,17 @@ dd:=divlog cc
--R a
--R Type: Expression
Integer
--E
+
+--S 74 14:200 Schaums and Axiom differ by a constant
+ee:=complexNormalize dd
+--R
+--R log(- 1)
+--R (5) - --------
+--R 3
+--R a
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.201~~~~~$\displaystyle
@@ -1019,7 +1091,7 @@ $$
<<*>>=
)clear all
---S 68
+--S 75
aa:=integrate(1/(x^2*(x^2+a^2)^(3/2)),x)
--R
--R
@@ -1031,7 +1103,7 @@ aa:=integrate(1/(x^2*(x^2+a^2)^(3/2)),x)
--R Type: Union(Expression
Integer,...)
--E
---S 69
+--S 76
bb:=-sqrt(x^2+a^2)/(a^4*x)-x/(a^4*sqrt(x^2+a^2))
--R
--R 2 2
@@ -1043,7 +1115,7 @@ bb:=-sqrt(x^2+a^2)/(a^4*x)-x/(a^4*sqrt(x^2+a^2))
--R Type: Expression
Integer
--E
---S 70 14:201 Schaums and Axiom differ by a constant
+--S 77 14:201 Schaums and Axiom differ by a constant
cc:=aa-bb
--R
--R 2
@@ -1064,7 +1136,7 @@ $$
<<*>>=
)clear all
---S 71
+--S 78
aa:=integrate(1/(x^3*(x^2+a^2)^(3/2)),x)
--R
--R
@@ -1091,7 +1163,7 @@ aa:=integrate(1/(x^3*(x^2+a^2)^(3/2)),x)
--R Type: Union(Expression
Integer,...)
--E
---S 72
+--S 79
bb:=-1/(2*a^2*x^2*sqrt(x^2+a^2))-3/(2*a^4*sqrt(x^2+a^2))+3/(2*a^5)*log((a+sqrt(x^2+a^2))/x)
--R
--R +-------+
@@ -1106,7 +1178,7 @@
bb:=-1/(2*a^2*x^2*sqrt(x^2+a^2))-3/(2*a^4*sqrt(x^2+a^2))+3/(2*a^5)*log((a+sqrt(x
--R Type: Expression
Integer
--E
---S 73
+--S 80
cc:=aa-bb
--R
--R (3)
@@ -1121,19 +1193,10 @@ cc:=aa-bb
--R Type: Expression
Integer
--E
---S 74
-divlog:=rule(log(a/b) == log(a) - log(b))
---R
---R a
---R (4) log(-) == - log(b) + log(a)
---R b
---R Type: RewriteRule(Integer,Integer,Expression
Integer)
---E
-
---S 75 14:202 Axiom cannot simplify this expression
-dd:=divlog cc
+--S 81
+dd:=expandLog cc
--R
---R (5)
+--R (4)
--R +-------+ +-------+
--R | 2 2 | 2 2
--R - 3log(\|x + a + a) + 3log(\|x + a - x + a)
@@ -1146,6 +1209,16 @@ dd:=divlog cc
--R 2a
--R Type: Expression
Integer
--E
+
+--S 82 14:202 Schaums and Axiom differ by a constant
+ee:=complexNormalize dd
+--R
+--R 3log(- 1)
+--R (5) ---------
+--R 5
+--R 2a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.203~~~~~$\displaystyle\int{(x^2+a^2)^{3/2}}~dx$}
@@ -1156,7 +1229,7 @@ $$
<<*>>=
)clear all
---S 76
+--S 83
aa:=integrate((x^2+a^2)^(3/2),x)
--R
--R (1)
@@ -1177,7 +1250,7 @@ aa:=integrate((x^2+a^2)^(3/2),x)
--R Type: Union(Expression
Integer,...)
--E
---S 77
+--S 84
bb:=(x*(x^2+a^2)^(3/2))/4+(3*a^2*x*sqrt(x^2+a^2))/8+3/8*a^4*log(x+sqrt(x^2+a^2))
--R
--R +-------+ +-------+
@@ -1188,7 +1261,7 @@
bb:=(x*(x^2+a^2)^(3/2))/4+(3*a^2*x*sqrt(x^2+a^2))/8+3/8*a^4*log(x+sqrt(x^2+a^2))
--R Type: Expression
Integer
--E
---S 78 14:203 Axiom cannot simplify this expression
+--S 85
cc:=aa-bb
--R
--R +-------+ +-------+
@@ -1198,6 +1271,16 @@ cc:=aa-bb
--R 8
--R Type: Expression
Integer
--E
+
+--S 86 14:203 Schaums and Axiom differ by a constant
+dd:=complexNormalize cc
+--R
+--R 4 2
+--R 3a log(a )
+--R (4) - ----------
+--R 8
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.204~~~~~$\displaystyle\int{x(x^2+a^2)^{3/2}}~dx$}
@@ -1205,7 +1288,7 @@ $$\int{x(x^2+a^2)^{3/2}}=\frac{(x^2+a^2)^{5/2}}{5}$$
<<*>>=
)clear all
---S 79
+--S 87
aa:=integrate(x*(x^2+a^2)^(3/2),x)
--R
--R
@@ -1223,7 +1306,7 @@ aa:=integrate(x*(x^2+a^2)^(3/2),x)
--R Type: Union(Expression
Integer,...)
--E
---S 80
+--S 88
bb:=(x^2+a^2)^(5/2)/5
--R
--R +-------+
@@ -1234,7 +1317,7 @@ bb:=(x^2+a^2)^(5/2)/5
--R Type: Expression
Integer
--E
---S 81 14:204 Schaums and Axiom agree
+--S 89 14:204 Schaums and Axiom agree
cc:=aa-bb
--R
--R (3) 0
@@ -1251,7 +1334,7 @@ $$
<<*>>=
)clear all
---S 82
+--S 90
aa:=integrate(x^2*(x^2+a^2)^(3/2),x)
--R
--R
@@ -1280,7 +1363,7 @@ aa:=integrate(x^2*(x^2+a^2)^(3/2),x)
--R Type: Union(Expression
Integer,...)
--E
---S 83
+--S 91
bb:=(x*(x^2+a^2)^(5/2))/6-(a^2*x*(x^2+a^2)^(3/2))/24-(a^4*x*sqrt(x^2+a^2))/16-a^6/16*log(x+sqrt(x^2+a^2))
--R
--R +-------+ +-------+
@@ -1291,7 +1374,7 @@
bb:=(x*(x^2+a^2)^(5/2))/6-(a^2*x*(x^2+a^2)^(3/2))/24-(a^4*x*sqrt(x^2+a^2))/16-a^
--R Type: Expression
Integer
--E
---S 84 14:205 Axiom cannot simplify this expression
+--S 92
cc:=aa-bb
--R
--R +-------+ +-------+
@@ -1301,6 +1384,16 @@ cc:=aa-bb
--R 16
--R Type: Expression
Integer
--E
+
+--S 93 14:205 Schaums and Axiom differ by a constant
+dd:=complexNormalize cc
+--R
+--R 6 2
+--R a log(a )
+--R (4) ---------
+--R 16
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.206~~~~~$\displaystyle\int{x^3(x^2+a^2)^{3/2}}~dx$}
@@ -1310,7 +1403,7 @@ $$
<<*>>=
)clear all
---S 85
+--S 94
aa:=integrate(x^3*(x^2+a^2)^(3/2),x)
--R
--R
@@ -1340,7 +1433,7 @@ aa:=integrate(x^3*(x^2+a^2)^(3/2),x)
--R Type: Union(Expression
Integer,...)
--E
---S 86
+--S 95
bb:=(x^2+a^2)^(7/2)/7-(a^2*(x^2+a^2)^(5/2))/5
--R
--R +-------+
@@ -1351,7 +1444,7 @@ bb:=(x^2+a^2)^(7/2)/7-(a^2*(x^2+a^2)^(5/2))/5
--R Type: Expression
Integer
--E
---S 87 14:206 Schaums and Axiom agree
+--S 96 14:206 Schaums and Axiom agree
cc:=aa-bb
--R
--R (3) 0
@@ -1368,7 +1461,7 @@ $$
<<*>>=
)clear all
---S 88
+--S 97
aa:=integrate((x^2+a^2)^(3/2)/x,x)
--R
--R
@@ -1391,7 +1484,7 @@ aa:=integrate((x^2+a^2)^(3/2)/x,x)
--R Type: Union(Expression
Integer,...)
--E
---S 89
+--S 98
bb:=(x^2+a^2)^(3/2)/3+a^2*sqrt(x^2+a^2)-a^3*log((a+sqrt(x^2+a^2))/x)
--R
--R +-------+
@@ -1404,7 +1497,7 @@
bb:=(x^2+a^2)^(3/2)/3+a^2*sqrt(x^2+a^2)-a^3*log((a+sqrt(x^2+a^2))/x)
--R Type: Expression
Integer
--E
---S 90
+--S 99
cc:=aa-bb
--R
--R (3)
@@ -1420,19 +1513,10 @@ cc:=aa-bb
--R Type: Expression
Integer
--E
---S 91
-divlog:=rule(log(a/b) == log(a) - log(b))
+--S 100
+dd:=expandLog cc
--R
---R a
---R (4) log(-) == - log(b) + log(a)
---R b
---R Type: RewriteRule(Integer,Integer,Expression
Integer)
---E
-
---S 92 14:207 Axiom cannot simplify this expression
-dd:=divlog cc
---R
---R (5)
+--R (4)
--R +-------+ +-------+
--R 3 | 2 2 3 | 2 2
--R a log(\|x + a + a) - a log(\|x + a - x + a)
@@ -1442,6 +1526,14 @@ dd:=divlog cc
--R a log(\|x + a - x - a) - a log(x)
--R Type: Expression
Integer
--E
+
+--S 101 14:207 Schaums and Axiom differ by a constant
+ee:=complexNormalize dd
+--R
+--R 3
+--R (5) - a log(- 1)
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.208~~~~~$\displaystyle
@@ -1453,7 +1545,7 @@ $$
<<*>>=
)clear all
---S 93
+--S 102
aa:=integrate((x^2+a^2)^{3/2}/x^2,x)
--R
--R
@@ -1472,7 +1564,7 @@ aa:=integrate((x^2+a^2)^{3/2}/x^2,x)
--R Type: Union(Expression
Integer,...)
--E
---S 94
+--S 103
bb:=-(x^2+a^2)^(3/2)/x+(3*x*sqrt(x^2+a^2))/2+3/2*a^2*log(x+sqrt(x^2+a^2))
--R
--R +-------+ +-------+
@@ -1483,7 +1575,7 @@
bb:=-(x^2+a^2)^(3/2)/x+(3*x*sqrt(x^2+a^2))/2+3/2*a^2*log(x+sqrt(x^2+a^2))
--R Type: Expression
Integer
--E
---S 95 14:208 Axiom cannot simplify this expression
+--S 104
cc:=aa-bb
--R
--R +-------+ +-------+
@@ -1493,6 +1585,17 @@ cc:=aa-bb
--R 2
--R Type: Expression
Integer
--E
+
+--S 105 14:208 Schaums and Axiom differ by a constant
+dd:=complexNormalize cc
+--R
+--R 2 2 2
+--R - 3a log(a ) - 2a
+--R (4) ------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.209~~~~~$\displaystyle
@@ -1504,7 +1607,7 @@ $$
<<*>>=
)clear all
---S 96
+--S 106
aa:=integrate((x^2+a^2)^(3/2)/x^3,x)
--R
--R
@@ -1527,48 +1630,60 @@ aa:=integrate((x^2+a^2)^(3/2)/x^3,x)
--R Type: Union(Expression
Integer,...)
--E
---S 97
-bb:=-(x^2+a^2)^(3/2)/(2*x^3)+3/2*sqrt(x^2+a^2)-3/2*a*log((a+sqrt(x^2+a^2))/x)
+--S 107
+bb:=-(x^2+a^2)^(3/2)/(2*x^2)+3/2*sqrt(x^2+a^2)-3/2*a*log((a+sqrt(x^2+a^2))/x)
--R
--R +-------+
---R | 2 2 +-------+
---R 3 \|x + a + a 3 2 2 | 2 2
---R - 3a x log(--------------) + (3x - x - a )\|x + a
+--R | 2 2 +-------+
+--R 2 \|x + a + a 2 2 | 2 2
+--R - 3a x log(--------------) + (2x - a )\|x + a
--R x
---R (2) ------------------------------------------------------
---R 3
---R 2x
+--R (2) -------------------------------------------------
+--R 2
+--R 2x
--R Type: Expression
Integer
--E
---S 98 14:209 Axiom cannot simplify this expression
+--S 108
cc:=aa-bb
--R
--R (3)
---R +-------+ +-------+
---R 5 3 3 | 2 2 6 3 4 | 2 2
---R ((- 12a x - 3a x )\|x + a + 12a x + 9a x )log(\|x + a - x +
a)
---R +
---R +-------+ +-------+
---R 5 3 3 | 2 2 6 3 4 | 2 2
---R ((12a x + 3a x )\|x + a - 12a x - 9a x )log(\|x + a - x - a)
---R +
---R +-------+
---R +-------+ | 2 2
---R 5 3 3 | 2 2 6 3 4 \|x + a + a
---R ((12a x + 3a x )\|x + a - 12a x - 9a x )log(--------------)
---R x
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R - 3a log(\|x + a - x + a) + 3a log(\|x + a - x - a)
--R +
---R +-------+
---R 6 5 2 4 2 3 4 2 4 | 2 2 7 6
2 5
---R (4x - 4x + 7a x - 7a x + 3a x - 3a x)\|x + a - 4x + 4x -
9a x
+--R +-------+
+--R | 2 2
+--R \|x + a + a
+--R 3a log(--------------)
+--R x
+--R /
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 109
+dd:=expandLog cc
+--R
+--R (4)
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R 3a log(\|x + a + a) - 3a log(\|x + a - x + a)
--R +
---R 2 4 4 3 4 2 6 6
---R 9a x - 6a x + 6a x - a x + a
+--R +-------+
+--R | 2 2
+--R 3a log(\|x + a - x - a) - 3a log(x)
--R /
---R +-------+
---R 5 2 3 | 2 2 6 2 4
---R (8x + 2a x )\|x + a - 8x - 6a x
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 110 14:209 Schaums and Axiom differ by a constant
+ee:=complexNormalize dd
+--R
+--R 3a log(- 1)
+--R (5) - -----------
+--R 2
--R Type: Expression
Integer
--E
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- [Axiom-developer] 20080418.01.tpd.patch (CATS Schaums-Axiom equivalence testing (1-11)),
daly <=