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[Axiom-developer] 20080421.01.tpd.patch (CATS Schaum2-Axiom equivalence
|
From: |
daly |
|
Subject: |
[Axiom-developer] 20080421.01.tpd.patch (CATS Schaum2-Axiom equivalence testing (14-16)) |
|
Date: |
Tue, 22 Apr 2008 00:56:31 -0500 |
schaum14.input.pamphlet
14:299 Schaums and Axiom agree
14:300 Schaums and Axiom agree
14:301 Schaums and Axiom agree
14:302 Schaums and Axiom agree
14:303 Schaums and Axiom agree
14:304 Schaums and Axiom agree
14:305 Schaums and Axiom agree
14:306 Schaums and Axiom agree
14:307 Schaums and Axiom agree
14:308 Schaums and Axiom agree
14:309 Axiom cannot compute this integral
14:310 Axiom cannot compute this integral
schaum15.input.pamphlet
14:311 Schaums and Axiom differ by a constant
14:312 Schaums and Axiom agree
14:313 Schaums and Axiom differ by a constant
14:314 Schaums and Axiom agree
14:315 Schaums and Axiom agree
14:316 Schaums and Axiom differ by a constant
14:317 Schaums and Axiom agree
14:318 Schaums and Axiom agree
14:319 Schaums and Axiom agree
14:320 Schaums and Axiom agree
14:321 Schaums and Axiom agree
14:322 Schaums and Axiom agree
14:323 Schaums and Axiom agree
14:324 Schaums and Axiom agree
schaum16.input.pamphlet
14:325 Axiom cannot simplify this expression
14:326 Schaums and Axiom agree
14:327 Axiom cannot compute this integral
14:328 Axiom cannot compute this integral
14:329 Schaums and Axiom differ by a constant
14:330 Schaums and Axiom agree
14:331 Schaums and Axiom agree
14:332 Axiom cannot compute this integral
14:333 Axiom cannot compute this integral
14:334 Axiom cannot simplify this expression
14:335 Axiom cannot compute this integral
14:336 Axiom cannot compute this integral
14:337 Axiom cannot compute this integral
14:338 Axiom cannot compute this integral
======================================================================
diff --git a/changelog b/changelog
index 29750e1..4b454b8 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,6 @@
+20080421 tpd src/input/schaum16.input show Schaums-Axiom equivalence
+20080421 tpd src/input/schaum15.input show Schaums-Axiom equivalence
+20080421 tpd src/input/schaum14.input show Schaums-Axiom equivalence
20080420 tpd src/input/schaum13.input show Schaums-Axiom equivalence
20080419 tpd src/input/schaum12.input show Schaums-Axiom equivalence
20080418 tpd src/input/schaum11.input show Schaums-Axiom equivalence
diff --git a/src/input/schaum14.input.pamphlet
b/src/input/schaum14.input.pamphlet
index 532a38a..3452b30 100644
--- a/src/input/schaum14.input.pamphlet
+++ b/src/input/schaum14.input.pamphlet
@@ -19,7 +19,7 @@ $$
)set message auto off
)clear all
---S 1 of 12
+--S 1
aa:=integrate(1/(x^3+a^3),x)
--R
--R
@@ -32,6 +32,41 @@ aa:=integrate(1/(x^3+a^3),x)
--R 6a \|3
--R Type: Union(Expression
Integer,...)
--E
+
+--S 2
+bb:=1/(6*a^2)*log((x+a)^2/(x^2-a*x+a^2))+1/(a^2*sqrt(3))*atan((2*x-a)/(a*sqrt(3)))
+--R
+--R 2 2 +-+
+--R x + 2a x + a +-+ (2x - a)\|3
+--R log(--------------) + 2\|3 atan(------------)
+--R 2 2 3a
+--R x - a x + a
+--R (2) ---------------------------------------------
+--R 2
+--R 6a
+--R Type: Expression
Integer
+--E
+
+--S 3
+cc:=aa-bb
+--R
+--R 2 2
+--R 2 2 x + 2a x + a
+--R - log(x - a x + a ) + 2log(x + a) - log(--------------)
+--R 2 2
+--R x - a x + a
+--R (3) --------------------------------------------------------
+--R 2
+--R 6a
+--R Type: Expression
Integer
+--E
+
+--S 4 14:299 Schaums and Axiom agree
+dd:=expandLog cc
+--R
+--R (4) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.300~~~~~$\displaystyle
@@ -43,7 +78,7 @@ $$
<<*>>=
)clear all
---S 2 of 12
+--S 5
aa:=integrate(x/(x^3+a^3),x)
--R
--R
@@ -56,6 +91,39 @@ aa:=integrate(x/(x^3+a^3),x)
--R 6a\|3
--R Type: Union(Expression
Integer,...)
--E
+
+--S 6
+bb:=1/(6*a)*log((x^2-a*x+a^2)/(x+a)^2)+1/(a*sqrt(3))*atan((2*x-a)/(a*sqrt(3)))
+--R
+--R 2 2 +-+
+--R x - a x + a +-+ (2x - a)\|3
+--R log(--------------) + 2\|3 atan(------------)
+--R 2 2 3a
+--R x + 2a x + a
+--R (2) ---------------------------------------------
+--R 6a
+--R Type: Expression
Integer
+--E
+
+--S 7
+cc:=aa-bb
+--R
+--R 2 2
+--R 2 2 x - a x + a
+--R log(x - a x + a ) - 2log(x + a) - log(--------------)
+--R 2 2
+--R x + 2a x + a
+--R (3) ------------------------------------------------------
+--R 6a
+--R Type: Expression
Integer
+--E
+
+--S 8 14:300 Schaums and Axiom agree
+dd:=expandLog cc
+--R
+--R (4) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.301~~~~~$\displaystyle
@@ -66,7 +134,7 @@ $$
<<*>>=
)clear all
---S 3 of 12
+--S 9
aa:=integrate(x^2/(x^3+a^3),x)
--R
--R
@@ -76,6 +144,23 @@ aa:=integrate(x^2/(x^3+a^3),x)
--R 3
--R Type: Union(Expression
Integer,...)
--E
+
+--S 10
+bb:=1/3*log(x^3+a^3)
+--R
+--R 3 3
+--R log(x + a )
+--R (2) ------------
+--R 3
+--R Type: Expression
Integer
+--E
+
+--S 11 14:301 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.302~~~~~$\displaystyle
@@ -86,7 +171,7 @@ $$
<<*>>=
)clear all
---S 4 of 12
+--S 12
aa:=integrate(1/(x*(x^3+a^3)),x)
--R
--R
@@ -97,6 +182,41 @@ aa:=integrate(1/(x*(x^3+a^3)),x)
--R 3a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 13
+bb:=1/(3*a^3)*log(x^3/(x^3+a^3))
+--R
+--R 3
+--R x
+--R log(-------)
+--R 3 3
+--R x + a
+--R (2) ------------
+--R 3
+--R 3a
+--R Type: Expression
Integer
+--E
+
+--S 14
+cc:=aa-bb
+--R
+--R 3
+--R 3 3 x
+--R - log(x + a ) + 3log(x) - log(-------)
+--R 3 3
+--R x + a
+--R (3) ---------------------------------------
+--R 3
+--R 3a
+--R Type: Expression
Integer
+--E
+
+--S 15 14:302 Schaums and Axiom agree
+dd:=expandLog cc
+--R
+--R (4) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.303~~~~~$\displaystyle
@@ -108,7 +228,7 @@ $$
<<*>>=
)clear all
---S 5 of 12
+--S 15
aa:=integrate(1/(x^2*(x^3+a^3)),x)
--R
--R
@@ -121,7 +241,42 @@ aa:=integrate(1/(x^2*(x^3+a^3)),x)
--R 4 +-+
--R 6a x\|3
--R Type: Union(Expression
Integer,...)
+--E
+
+--S 16
+bb:=-1/(a^3*x)-1/(6*a^4)*log((x^2-a*x+a^2)/(x+a)^2)-1/(a^4*sqrt(3))*atan((2*x-a)/(a*sqrt(3)))
+--R
+--R 2 2 +-+
+--R x - a x + a +-+ (2x - a)\|3
+--R - x log(--------------) - 2x\|3 atan(------------) - 6a
+--R 2 2 3a
+--R x + 2a x + a
+--R (2) -------------------------------------------------------
+--R 4
+--R 6a x
+--R Type: Expression
Integer
--E
+
+--S 17
+cc:=aa-bb
+--R
+--R 2 2
+--R 2 2 x - a x + a
+--R - log(x - a x + a ) + 2log(x + a) + log(--------------)
+--R 2 2
+--R x + 2a x + a
+--R (3) --------------------------------------------------------
+--R 4
+--R 6a
+--R Type: Expression
Integer
+--E
+
+--S 18 14:303 Schaums and Axiom agree
+dd:=expandLog cc
+--R
+--R (4) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.304~~~~~$\displaystyle
@@ -134,7 +289,7 @@ $$
<<*>>=
)clear all
---S 6 of 12
+--S 19
aa:=integrate(1/(x^3+a^3)^2,x)
--R
--R
@@ -151,6 +306,42 @@ aa:=integrate(1/(x^3+a^3)^2,x)
--R (9a x + 9a )\|3
--R Type: Union(Expression
Integer,...)
--E
+
+--S 20
+bb:=x/(3*a^3*(x^3+a^3))+1/(9*a^5)*log((x+a)^2/(x^2-a*x+a^2))+2/(3*a^5*sqrt(3))*atan((2*x-a)/(a*sqrt(3)))
+--R
+--R (2)
+--R 2 2 +-+
+--R 3 3 x + 2a x + a 3 3 +-+ (2x - a)\|3 2
+--R (x + a )log(--------------) + (2x + 2a )\|3 atan(------------) + 3a x
+--R 2 2 3a
+--R x - a x + a
+--R -----------------------------------------------------------------------
+--R 5 3 8
+--R 9a x + 9a
+--R Type: Expression
Integer
+--E
+
+--S 21
+cc:=aa-bb
+--R
+--R 2 2
+--R 2 2 x + 2a x + a
+--R - log(x - a x + a ) + 2log(x + a) - log(--------------)
+--R 2 2
+--R x - a x + a
+--R (3) --------------------------------------------------------
+--R 5
+--R 9a
+--R Type: Expression
Integer
+--E
+
+--S 22 14:304 Schaums and Axiom agree
+dd:=expandLog cc
+--R
+--R (4) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.305~~~~~$\displaystyle
@@ -163,7 +354,7 @@ $$
<<*>>=
)clear all
---S 7 of 12
+--S 23
aa:=integrate(x/(x^3+a^3)^2,x)
--R
--R
@@ -180,6 +371,42 @@ aa:=integrate(x/(x^3+a^3)^2,x)
--R (18a x + 18a )\|3
--R Type: Union(Expression
Integer,...)
--E
+
+--S 24
+bb:=x^2/(3*a^3*(x^3+a^3))+1/(18*a^4)*log((x^2-a*x+a^2)/(x+a)^2)+1/(3*a^4*sqrt(3))*atan((2*x-a)/(a*sqrt(3)))
+--R
+--R (2)
+--R 2 2 +-+
+--R 3 3 x - a x + a 3 3 +-+ (2x - a)\|3 2
+--R (x + a )log(--------------) + (2x + 2a )\|3 atan(------------) + 6a x
+--R 2 2 3a
+--R x + 2a x + a
+--R ------------------------------------------------------------------------
+--R 4 3 7
+--R 18a x + 18a
+--R Type: Expression
Integer
+--E
+
+--S 25
+cc:=aa-bb
+--R
+--R 2 2
+--R 2 2 x - a x + a
+--R log(x - a x + a ) - 2log(x + a) - log(--------------)
+--R 2 2
+--R x + 2a x + a
+--R (3) ------------------------------------------------------
+--R 4
+--R 18a
+--R Type: Expression
Integer
+--E
+
+--S 26 14:305 Schaums and Axiom agree
+dd:=expandLog cc
+--R
+--R (4) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.306~~~~~$\displaystyle
@@ -190,7 +417,7 @@ $$
<<*>>=
)clear all
---S 8 of 12
+--S 27
aa:=integrate(x^2/(x^3+a^3)^2,x)
--R
--R
@@ -199,7 +426,24 @@ aa:=integrate(x^2/(x^3+a^3)^2,x)
--R 3 3
--R 3x + 3a
--R Type: Union(Expression
Integer,...)
+--E
+
+--S 28
+bb:=-1/(3*(x^3+a^3))
+--R
+--R 1
+--R (2) - ---------
+--R 3 3
+--R 3x + 3a
+--R Type: Fraction Polynomial
Integer
--E
+
+--S 29 14:306 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.307~~~~~$\displaystyle
@@ -210,7 +454,7 @@ $$
<<*>>=
)clear all
---S 9 of 12
+--S 30
aa:=integrate(1/(x*(x^3+a^3)^2),x)
--R
--R
@@ -221,6 +465,41 @@ aa:=integrate(1/(x*(x^3+a^3)^2),x)
--R 3a x + 3a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 31
+bb:=1/(3*a^3*(x^3+a^3))+1/(3*a^6)*log(x^3/(x^3+a^3))
+--R
+--R 3
+--R 3 3 x 3
+--R (x + a )log(-------) + a
+--R 3 3
+--R x + a
+--R (2) --------------------------
+--R 6 3 9
+--R 3a x + 3a
+--R Type: Expression
Integer
+--E
+
+--S 32
+cc:=aa-bb
+--R
+--R 3
+--R 3 3 x
+--R - log(x + a ) + 3log(x) - log(-------)
+--R 3 3
+--R x + a
+--R (3) ---------------------------------------
+--R 6
+--R 3a
+--R Type: Expression
Integer
+--E
+
+--S 33 14:307 Schaums and Axiom agree
+dd:=expandLog cc
+--R
+--R (4) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.308~~~~~$\displaystyle
@@ -232,7 +511,7 @@ $$
<<*>>=
)clear all
---S 10 of 12
+--S 34
aa:=integrate(1/(x^2*(x^3+a^3)^2),x)
--R
--R
@@ -248,7 +527,44 @@ aa:=integrate(1/(x^2*(x^3+a^3)^2),x)
--R 7 4 10 +-+
--R (9a x + 9a x)\|3
--R Type: Union(Expression
Integer,...)
+--E
+
+--S 35
+t1:=integrate(x/(x^3+a^3),x)
+--R
+--R +-+
+--R +-+ 2 2 +-+ (2x - a)\|3
+--R \|3 log(x - a x + a ) - 2\|3 log(x + a) + 6atan(------------)
+--R 3a
+--R (2) --------------------------------------------------------------
+--R +-+
+--R 6a\|3
+--R Type: Union(Expression
Integer,...)
+--E
+
+--S 36
+bb:=-1/(a^6*x)-x^2/(3*a^6*(x^3+a^3))-4/(3*a^6)*t1
+--R
+--R (3)
+--R 4 3 +-+ 2 2 4 3 +-+
+--R (- 2x - 2a x)\|3 log(x - a x + a ) + (4x + 4a x)\|3 log(x + a)
+--R +
+--R +-+
+--R 4 3 (2x - a)\|3 3 4 +-+
+--R (- 12x - 12a x)atan(------------) + (- 12a x - 9a )\|3
+--R 3a
+--R /
+--R 7 4 10 +-+
+--R (9a x + 9a x)\|3
+--R Type: Expression
Integer
--E
+
+--S 37 14:308 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (4) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.309~~~~~$\displaystyle
@@ -259,7 +575,7 @@ $$
<<*>>=
)clear all
---S 11 of 12
+--S 38 14:309 Axiom cannot compute this integral
aa:=integrate(x^m/(x^3+a^3),x)
--R
--R
@@ -269,7 +585,7 @@ aa:=integrate(x^m/(x^3+a^3),x)
--R ++ 3 3
--I a + %L
--R Type: Union(Expression
Integer,...)
---E
+--E
@
\section{\cite{1}:14.310~~~~~$\displaystyle
@@ -280,7 +596,7 @@ $$
<<*>>=
)clear all
---S 12 of 12
+--S 39 14:310 Axiom cannot compute this integral
aa:=integrate(1/(x^n*(x^3+a^3)),x)
--R
--R
diff --git a/src/input/schaum15.input.pamphlet
b/src/input/schaum15.input.pamphlet
index 51a6094..0c01fbb 100644
--- a/src/input/schaum15.input.pamphlet
+++ b/src/input/schaum15.input.pamphlet
@@ -20,7 +20,7 @@ $$
)set message auto off
)clear all
---S 1 of 14
+--S 1
aa:=integrate(1/(x^4+a^4),x)
--R
--R
@@ -53,6 +53,202 @@ aa:=integrate(1/(x^4+a^4),x)
--R \|2
--R Type: Union(Expression
Integer,...)
--E
+
+--S 2
+bb:=1/(4*a^3*sqrt(2))*log((x^2+a*x*sqrt(2)+a^2)/(x^2-a*x*sqrt(2)+a^2))-1/(2*a^3*sqrt(2))*atan((a*x*sqrt(2))/(x^2-a^2))
+--R
+--R +-+ 2 2 +-+
+--R +-+ - a x\|2 - x - a +-+ a x\|2
+--R \|2 log(-------------------) - 2\|2 atan(-------)
+--R +-+ 2 2 2 2
+--R a x\|2 - x - a x - a
+--R (2) -------------------------------------------------
+--R 3
+--R 8a
+--R Type: Expression
Integer
+--E
+
+--S 3
+cc:=aa-bb
+--R
+--R (3)
+--R +------+ +------+2 +------+
+--R 3 | 1 8 | 1 4 +-+ | 1 2
+--R 4a |------ log(16a |------ + 4a x\|2 |------ + x )
+--R 4| 12 4| 12 4| 12
+--R \|256a \|256a \|256a
+--R +
+--R +------+ +------+2 +------+
+--R 3 | 1 8 | 1 4 +-+ | 1 2
+--R - 4a |------ log(16a |------ - 4a x\|2 |------ + x )
+--R 4| 12 4| 12 4| 12
+--R \|256a \|256a \|256a
+--R +
+--R +------+
+--R 4 | 1
+--R 4a |------
+--R +------+ 4| 12
+--R 3 | 1 \|256a
+--R 8a |------ atan(--------------------)
+--R 4| 12 +------+
+--R \|256a 4 | 1 +-+
+--R 4a |------ - x\|2
+--R 4| 12
+--R \|256a
+--R +
+--R +------+
+--R 4 | 1
+--R 4a |------
+--R +------+ 4| 12 +-+ 2 2
+--R 3 | 1 \|256a - a x\|2 - x - a
+--R - 8a |------ atan(-------------------- - log(-------------------)
+--R 4| 12 +------+ +-+ 2 2
+--R \|256a 4 | 1 +-+ a x\|2 - x - a
+--R 4a |------ + x\|2
+--R 4| 12
+--R \|256a
+--R +
+--R +-+
+--R a x\|2
+--R 2atan(-------)
+--R 2 2
+--R x - a
+--R /
+--R 3 +-+
+--R 4a \|2
+--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 (5)
+--R +------+ +------+2 +------+
+--R 3 | 1 8 | 1 4 +-+ | 1 2
+--R 4a |------ log(16a |------ + 4a x\|2 |------ + x )
+--R 4| 12 4| 12 4| 12
+--R \|256a \|256a \|256a
+--R +
+--R +------+ +------+2 +------+
+--R 3 | 1 8 | 1 4 +-+ | 1 2
+--R - 4a |------ log(16a |------ - 4a x\|2 |------ + x )
+--R 4| 12 4| 12 4| 12
+--R \|256a \|256a \|256a
+--R +
+--R +------+
+--R 4 | 1 +-+
+--R (- 4 + 4%i)a |------ + %i x\|2
+--R +------+ 4| 12
+--R 3 | 1 \|256a
+--R 4%i a |------ log(---------------------------------)
+--R 4| 12 +------+
+--R \|256a 4 | 1 +-+
+--R (4 + 4%i)a |------ + %i x\|2
+--R 4| 12
+--R \|256a
+--R +
+--R +------+
+--R 4 | 1 +-+
+--R (- 4 + 4%i)a |------ - %i x\|2
+--R +------+ 4| 12
+--R 3 | 1 \|256a
+--R - 4%i a |------ log(---------------------------------)
+--R 4| 12 +------+
+--R \|256a 4 | 1 +-+
+--R (4 + 4%i)a |------ - %i x\|2
+--R 4| 12
+--R \|256a
+--R +
+--R +-+ 2 2 +-+ 2 2
+--R - a x\|2 + %i x - %i a - a x\|2 - x - a
+--R - %i log(-------------------------) - log(-------------------)
+--R +-+ 2 2 +-+ 2 2
+--R a x\|2 + %i x - %i a a x\|2 - x - a
+--R /
+--R 3 +-+
+--R 4a \|2
+--R Type: Expression Complex
Integer
+--E
+
+--S 6
+ee:=rootSimp dd
+--R
+--R (6)
+--R +-+
+--R +-+ 2 2 x\|2 + (1 + %i)a
+--R log(a x\|2 + x + a ) + %i log(-----------------)
+--R +-+
+--R x\|2 + (1 - %i)a
+--R +
+--R +-+ +-+ 2 2
+--R x\|2 + (- 1 - %i)a - a x\|2 + %i x - %i a
+--R - %i log(-------------------) - %i log(-------------------------)
+--R +-+ +-+ 2 2
+--R x\|2 + (- 1 + %i)a a x\|2 + %i x - %i a
+--R +
+--R +-+ 2 2
+--R - a x\|2 - x - a +-+ 2 2
+--R - log(-------------------) - log(- a x\|2 + x + a )
+--R +-+ 2 2
+--R a x\|2 - x - a
+--R /
+--R 3 +-+
+--R 4a \|2
+--R Type: Expression Complex
Integer
+--E
+
+--S 7
+ff:=expandLog ee
+--R
+--R (7)
+--R +-+ 2 2 +-+ 2 2
+--R %i log(a x\|2 + %i x - %i a ) - %i log(a x\|2 - %i x + %i a )
+--R +
+--R +-+ +-+
+--R %i log(x\|2 + (1 + %i)a) - %i log(x\|2 + (1 - %i)a)
+--R +
+--R +-+ +-+
+--R %i log(x\|2 + (- 1 + %i)a) - %i log(x\|2 + (- 1 - %i)a)
+--R +
+--R (- 2 - %i)log(- 1)
+--R /
+--R 3 +-+
+--R 4a \|2
+--R Type: Expression Complex
Integer
+--E
+
+--S 8
+gg:=complexNormalize ff
+--R
+--R %i %i
+--R %i log(--) - %i log(- --) + (- 2 - %i)log(- 1)
+--R 2 2
+--R (8) ----------------------------------------------
+--R 3 +-+
+--R 4a \|2
+--R Type: Expression Complex
Integer
+--E
+
+--S 9 14:311 Schaums and Axiom differ by a constant
+hh:=expandLog gg
+--R
+--R %i log(%i) - %i log(- %i) + (- 2 - %i)log(- 1)
+--R (9) ----------------------------------------------
+--R 3 +-+
+--R 4a \|2
+--R Type: Expression Complex
Integer
+--E
@
\section{\cite{1}:14.312~~~~~$\displaystyle
@@ -63,7 +259,7 @@ $$
<<*>>=
)clear all
---S 2 of 14
+--S 10
aa:=integrate(x/(x^4+a^4),x)
--R
--R
@@ -77,6 +273,27 @@ aa:=integrate(x/(x^4+a^4),x)
--R 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 11
+bb:=1/(2*a^2)*atan(x^2/a^2)
+--R
+--R 2
+--R x
+--R atan(--)
+--R 2
+--R a
+--R (2) --------
+--R 2
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 12 14:312 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.313~~~~~$\displaystyle
@@ -89,7 +306,7 @@ $$
<<*>>=
)clear all
---S 3 of 14
+--S 13
aa:=integrate(x^2/(x^4+a^4),x)
--R
--R
@@ -122,6 +339,213 @@ aa:=integrate(x^2/(x^4+a^4),x)
--R \|2
--R Type: Union(Expression
Integer,...)
--E
+
+--S 14
+bb:=1/(4*a*sqrt(2))*log((x^2-a*x*sqrt(2)+a^2)/(x^2+a*x*sqrt(2)+a^2))-1/(2*a*sqrt(2))*atan((a*x*sqrt(2))/(x^2-a^2))
+--R
+--R +-+ 2 2 +-+
+--R +-+ - a x\|2 + x + a +-+ a x\|2
+--R \|2 log(-------------------) - 2\|2 atan(-------)
+--R +-+ 2 2 2 2
+--R a x\|2 + x + a x - a
+--R (2) -------------------------------------------------
+--R 8a
+--R Type: Expression
Integer
+--E
+
+--S 15
+cc:=aa-bb
+--R
+--R (3)
+--R +-----+ +-----+3 +-----+2
+--R | 1 4 +-+ | 1 4 | 1 2
+--R - 4a |----- log(64a x\|2 |----- + 16a |----- + x )
+--R 4| 4 4| 4 4| 4
+--R \|256a \|256a \|256a
+--R +
+--R +-----+ +-----+3 +-----+2
+--R | 1 4 +-+ | 1 4 | 1 2
+--R 4a |----- log(- 64a x\|2 |----- + 16a |----- + x )
+--R 4| 4 4| 4 4| 4
+--R \|256a \|256a \|256a
+--R +
+--R +-----+3
+--R 4 | 1
+--R 64a |-----
+--R +-----+ 4| 4
+--R | 1 \|256a
+--R 8a |----- atan(---------------------)
+--R 4| 4 +-----+3
+--R \|256a 4 | 1 +-+
+--R 64a |----- - x\|2
+--R 4| 4
+--R \|256a
+--R +
+--R +-----+3
+--R 4 | 1
+--R 64a |-----
+--R +-----+ 4| 4 +-+ 2 2
+--R | 1 \|256a - a x\|2 + x + a
+--R - 8a |----- atan(--------------------- - log(-------------------)
+--R 4| 4 +-----+3 +-+ 2 2
+--R \|256a 4 | 1 +-+ a x\|2 + x + a
+--R 64a |----- + x\|2
+--R 4| 4
+--R \|256a
+--R +
+--R +-+
+--R a x\|2
+--R 2atan(-------)
+--R 2 2
+--R x - a
+--R /
+--R +-+
+--R 4a\|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 (5)
+--R +-----+ +-----+3 +-----+2
+--R | 1 4 +-+ | 1 4 | 1 2
+--R - 4a |----- log(64a x\|2 |----- + 16a |----- + x )
+--R 4| 4 4| 4 4| 4
+--R \|256a \|256a \|256a
+--R +
+--R +-----+3
+--R 4 | 1 +-+
+--R (- 64 + 64%i)a |----- + %i x\|2
+--R +-----+ 4| 4
+--R | 1 \|256a
+--R 4%i a |----- log(-----------------------------------)
+--R 4| 4 +-----+3
+--R \|256a 4 | 1 +-+
+--R (64 + 64%i)a |----- + %i x\|2
+--R 4| 4
+--R \|256a
+--R +
+--R +-----+3
+--R 4 | 1 +-+
+--R (- 64 + 64%i)a |----- - %i x\|2
+--R +-----+ 4| 4
+--R | 1 \|256a
+--R - 4%i a |----- log(-----------------------------------)
+--R 4| 4 +-----+3
+--R \|256a 4 | 1 +-+
+--R (64 + 64%i)a |----- - %i x\|2
+--R 4| 4
+--R \|256a
+--R +
+--R +-----+ +-----+3 +-----+2
+--R | 1 4 +-+ | 1 4 | 1 2
+--R 4a |----- log(- 64a x\|2 |----- + 16a |----- + x )
+--R 4| 4 4| 4 4| 4
+--R \|256a \|256a \|256a
+--R +
+--R +-+ 2 2 +-+ 2 2
+--R - a x\|2 + x + a - a x\|2 + %i x - %i a
+--R - log(-------------------) - %i log(-------------------------)
+--R +-+ 2 2 +-+ 2 2
+--R a x\|2 + x + a a x\|2 + %i x - %i a
+--R /
+--R +-+
+--R 4a\|2
+--R Type: Expression Complex
Integer
+--E
+
+--S 18
+ee:=expandLog dd
+--R
+--R (6)
+--R +-----+ +-----+3 +-----+2
+--R | 1 4 +-+ | 1 4 | 1 2
+--R - 4a |----- log(64a x\|2 |----- + 16a |----- + x )
+--R 4| 4 4| 4 4| 4
+--R \|256a \|256a \|256a
+--R +
+--R +-----+ +-----+3 +-----+2
+--R | 1 4 +-+ | 1 4 | 1 2
+--R 4a |----- log(64a x\|2 |----- - 16a |----- - x )
+--R 4| 4 4| 4 4| 4
+--R \|256a \|256a \|256a
+--R +
+--R +-----+ +-----+3
+--R | 1 4 | 1 +-+
+--R 4%i a |----- log((64 + 64%i)a |----- + x\|2 )
+--R 4| 4 4| 4
+--R \|256a \|256a
+--R +
+--R +-----+ +-----+3
+--R | 1 4 | 1 +-+
+--R - 4%i a |----- log((64 + 64%i)a |----- + %i x\|2 )
+--R 4| 4 4| 4
+--R \|256a \|256a
+--R +
+--R +-----+ +-----+3
+--R | 1 4 | 1 +-+
+--R 4%i a |----- log((64 + 64%i)a |----- - %i x\|2 )
+--R 4| 4 4| 4
+--R \|256a \|256a
+--R +
+--R +-----+ +-----+3
+-----+
+--R | 1 4 | 1 +-+ |
1
+--R - 4%i a |----- log((64 + 64%i)a |----- - x\|2 + 4a log(- 1)
|-----
+--R 4| 4 4| 4 4|
4
+--R \|256a \|256a
\|256a
+--R +
+--R +-+ 2 2 +-+ 2 2
+--R log(a x\|2 + x + a ) + %i log(a x\|2 + %i x - %i a )
+--R +
+--R +-+ 2 2 +-+ 2 2
+--R - %i log(a x\|2 - %i x + %i a ) - log(a x\|2 - x - a )
+--R +
+--R (- 1 - %i)log(- 1)
+--R /
+--R +-+
+--R 4a\|2
+--R Type: Expression Complex
Integer
+--E
+
+--S 19
+ff:=rootSimp ee
+--R
+--R (7)
+--R +-+ 2 2 +-+ 2 2
+--R %i log(a x\|2 + %i x - %i a ) - %i log(a x\|2 - %i x + %i a )
+--R +
+--R +-+ +-+
+--R %i log(x\|2 + (1 + %i)a) - %i log(%i x\|2 + (1 + %i)a)
+--R +
+--R +-+ +-+
+--R %i log(- %i x\|2 + (1 + %i)a) - %i log(- x\|2 + (1 + %i)a) - %i
log(- 1)
+--R /
+--R +-+
+--R 4a\|2
+--R Type: Expression Complex
Integer
+--E
+
+--S 20 14:313 Schaums and Axiom differ by a constant
+gg:=complexNormalize ff
+--R
+--R %i log(2) - %i log(- 1) - %i log(- 2)
+--R (8) -------------------------------------
+--R +-+
+--R 4a\|2
+--R Type: Expression Complex
Integer
+--E
@
\section{\cite{1}:14.314~~~~~$\displaystyle
@@ -132,7 +556,7 @@ $$
<<*>>=
)clear all
---S 4 of 14
+--S 21
aa:=integrate(x^3/(x^4+a^4),x)
--R
--R
@@ -141,7 +565,24 @@ aa:=integrate(x^3/(x^4+a^4),x)
--R (1) ------------
--R 4
--R Type: Union(Expression
Integer,...)
+--E
+
+--S 22
+bb:=1/4*log(x^4+a^4)
+--R
+--R 4 4
+--R log(x + a )
+--R (2) ------------
+--R 4
+--R Type: Expression
Integer
--E
+
+--S 23 14:314 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.315~~~~~$\displaystyle
@@ -152,7 +593,7 @@ $$
<<*>>=
)clear all
---S 5 of 14
+--S 24
aa:=integrate(1/(x*(x^4+a^4)),x)
--R
--R
@@ -163,6 +604,41 @@ aa:=integrate(1/(x*(x^4+a^4)),x)
--R 4a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 25
+bb:=1/(4*a^4)*log(x^4/(x^4+a^4))
+--R
+--R 4
+--R x
+--R log(-------)
+--R 4 4
+--R x + a
+--R (2) ------------
+--R 4
+--R 4a
+--R Type: Expression
Integer
+--E
+
+--S 26
+cc:=aa-bb
+--R
+--R 4
+--R 4 4 x
+--R - log(x + a ) + 4log(x) - log(-------)
+--R 4 4
+--R x + a
+--R (3) ---------------------------------------
+--R 4
+--R 4a
+--R Type: Expression
Integer
+--E
+
+--S 27 14:315 Schaums and Axiom agree
+dd:=expandLog cc
+--R
+--R (4) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.316~~~~~$\displaystyle
@@ -175,7 +651,7 @@ $$
<<*>>=
)clear all
---S 6 of 14
+--S 28
aa:=integrate(1/(x^2*(x^4+a^4)),x)
--R
--R
@@ -220,6 +696,220 @@ aa:=integrate(1/(x^2*(x^4+a^4)),x)
--R a x\|2
--R Type: Union(Expression
Integer,...)
--E
+
+--S 29
+bb:=-1/(a^4*x)-1/(4*a^5*sqrt(2))*log((x^2-a*x*sqrt(2)+a^2)/(x^2+a*x*sqrt(2)+a^2))+1/(2*a^5*sqrt(2))*atan((a*x*sqrt(2))/(x^2-a^2))
+--R
+--R +-+ 2 2 +-+
+--R +-+ - a x\|2 + x + a +-+ a x\|2
+--R - x\|2 log(-------------------) + 2x\|2 atan(-------) - 8a
+--R +-+ 2 2 2 2
+--R a x\|2 + x + a x - a
+--R (2) ----------------------------------------------------------
+--R 5
+--R 8a x
+--R Type: Expression
Integer
+--E
+
+--S 30
+cc:=aa-bb
+--R
+--R (3)
+--R +------+ +------+3 +------+2
+--R 5 | 1 16 +-+ | 1 12 | 1 2
+--R 4a |------ log(64a x\|2 |------ + 16a |------ + x )
+--R 4| 20 4| 20 4| 20
+--R \|256a \|256a \|256a
+--R +
+--R +------+ +------+3 +------+2
+--R 5 | 1 16 +-+ | 1 12 | 1 2
+--R - 4a |------ log(- 64a x\|2 |------ + 16a |------ + x )
+--R 4| 20 4| 20 4| 20
+--R \|256a \|256a \|256a
+--R +
+--R +------+3
+--R 16 | 1
+--R 64a |------
+--R +------+ 4| 20
+--R 5 | 1 \|256a
+--R - 8a |------ atan(-----------------------)
+--R 4| 20 +------+3
+--R \|256a 16 | 1 +-+
+--R 64a |------ - x\|2
+--R 4| 20
+--R \|256a
+--R +
+--R +------+3
+--R 16 | 1
+--R 64a |------
+--R +------+ 4| 20 +-+ 2 2
+--R 5 | 1 \|256a - a x\|2 + x + a
+--R 8a |------ atan(----------------------- + log(-------------------)
+--R 4| 20 +------+3 +-+ 2 2
+--R \|256a 16 | 1 +-+ a x\|2 + x + a
+--R 64a |------ + x\|2
+--R 4| 20
+--R \|256a
+--R +
+--R +-+
+--R a x\|2
+--R - 2atan(-------)
+--R 2 2
+--R x - a
+--R /
+--R 5 +-+
+--R 4a \|2
+--R Type: Expression
Integer
+--E
+
+--S 31
+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 32
+dd:=atanrule cc
+--R
+--R (5)
+--R +------+ +------+3 +------+2
+--R 5 | 1 16 +-+ | 1 12 | 1 2
+--R 4a |------ log(64a x\|2 |------ + 16a |------ + x )
+--R 4| 20 4| 20 4| 20
+--R \|256a \|256a \|256a
+--R +
+--R +------+3
+--R 16 | 1 +-+
+--R (- 64 + 64%i)a |------ + %i x\|2
+--R +------+ 4| 20
+--R 5 | 1 \|256a
+--R - 4%i a |------ log(-------------------------------------)
+--R 4| 20 +------+3
+--R \|256a 16 | 1 +-+
+--R (64 + 64%i)a |------ + %i x\|2
+--R 4| 20
+--R \|256a
+--R +
+--R +------+3
+--R 16 | 1 +-+
+--R (- 64 + 64%i)a |------ - %i x\|2
+--R +------+ 4| 20
+--R 5 | 1 \|256a
+--R 4%i a |------ log(-------------------------------------)
+--R 4| 20 +------+3
+--R \|256a 16 | 1 +-+
+--R (64 + 64%i)a |------ - %i x\|2
+--R 4| 20
+--R \|256a
+--R +
+--R +------+ +------+3 +------+2
+--R 5 | 1 16 +-+ | 1 12 | 1 2
+--R - 4a |------ log(- 64a x\|2 |------ + 16a |------ + x )
+--R 4| 20 4| 20 4| 20
+--R \|256a \|256a \|256a
+--R +
+--R +-+ 2 2 +-+ 2 2
+--R - a x\|2 + x + a - a x\|2 + %i x - %i a
+--R log(-------------------) + %i log(-------------------------)
+--R +-+ 2 2 +-+ 2 2
+--R a x\|2 + x + a a x\|2 + %i x - %i a
+--R /
+--R 5 +-+
+--R 4a \|2
+--R Type: Expression Complex
Integer
+--E
+
+--S 33
+ee:=expandLog dd
+--R
+--R (6)
+--R +------+ +------+3 +------+2
+--R 5 | 1 16 +-+ | 1 12 | 1 2
+--R 4a |------ log(64a x\|2 |------ + 16a |------ + x )
+--R 4| 20 4| 20 4| 20
+--R \|256a \|256a \|256a
+--R +
+--R +------+ +------+3 +------+2
+--R 5 | 1 16 +-+ | 1 12 | 1 2
+--R - 4a |------ log(64a x\|2 |------ - 16a |------ - x )
+--R 4| 20 4| 20 4| 20
+--R \|256a \|256a \|256a
+--R +
+--R +------+ +------+3
+--R 5 | 1 16 | 1 +-+
+--R - 4%i a |------ log((64 + 64%i)a |------ + x\|2 )
+--R 4| 20 4| 20
+--R \|256a \|256a
+--R +
+--R +------+ +------+3
+--R 5 | 1 16 | 1 +-+
+--R 4%i a |------ log((64 + 64%i)a |------ + %i x\|2 )
+--R 4| 20 4| 20
+--R \|256a \|256a
+--R +
+--R +------+ +------+3
+--R 5 | 1 16 | 1 +-+
+--R - 4%i a |------ log((64 + 64%i)a |------ - %i x\|2 )
+--R 4| 20 4| 20
+--R \|256a \|256a
+--R +
+--R +------+ +------+3
+--R 5 | 1 16 | 1 +-+
+--R 4%i a |------ log((64 + 64%i)a |------ - x\|2 )
+--R 4| 20 4| 20
+--R \|256a \|256a
+--R +
+--R +------+
+--R 5 | 1 +-+ 2 2
+--R - 4a log(- 1) |------ - log(a x\|2 + x + a )
+--R 4| 20
+--R \|256a
+--R +
+--R +-+ 2 2 +-+ 2 2
+--R - %i log(a x\|2 + %i x - %i a ) + %i log(a x\|2 - %i x + %i a )
+--R +
+--R +-+ 2 2
+--R log(a x\|2 - x - a ) + (1 + %i)log(- 1)
+--R /
+--R 5 +-+
+--R 4a \|2
+--R Type: Expression Complex
Integer
+--E
+
+--S 34
+ff:=rootSimp ee
+--R
+--R (7)
+--R +-+ 2 2 +-+ 2 2
+--R - %i log(a x\|2 + %i x - %i a ) + %i log(a x\|2 - %i x + %i a )
+--R +
+--R +-+ +-+
+--R - %i log(x\|2 + (1 + %i)a) + %i log(%i x\|2 + (1 + %i)a)
+--R +
+--R +-+ +-+
+--R - %i log(- %i x\|2 + (1 + %i)a) + %i log(- x\|2 + (1 + %i)a)
+--R +
+--R %i log(- 1)
+--R /
+--R 5 +-+
+--R 4a \|2
+--R Type: Expression Complex
Integer
+--E
+
+--S 35 14:316 Schaums and Axiom differ by a constant
+gg:=complexNormalize ff
+--R
+--R - %i log(2) + %i log(- 1) + %i log(- 2)
+--R (8) ---------------------------------------
+--R 5 +-+
+--R 4a \|2
+--R Type: Expression Complex
Integer
+--E
@
\section{\cite{1}:14.317~~~~~$\displaystyle
@@ -230,7 +920,7 @@ $$
<<*>>=
)clear all
---S 7 of 14
+--S 36
aa:=integrate(1/(x^3*(x^4+a^4)),x)
--R
--R
@@ -244,6 +934,27 @@ aa:=integrate(1/(x^3*(x^4+a^4)),x)
--R 2a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 37
+bb:=-1/(2*a^4*x^2)-1/(2*a^6)*atan(x^2/a^2)
+--R
+--R 2
+--R 2 x 2
+--R - x atan(--) - a
+--R 2
+--R a
+--R (2) -----------------
+--R 6 2
+--R 2a x
+--R Type: Expression
Integer
+--E
+
+--S 38 14:317 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.318~~~~~$\displaystyle
@@ -255,7 +966,7 @@ $$
<<*>>=
)clear all
---S 8 of 14
+--S 39
aa:=integrate(1/(x^4-a^4),x)
--R
--R
@@ -267,6 +978,37 @@ aa:=integrate(1/(x^4-a^4),x)
--R 4a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 40
+bb:=1/(4*a^3)*log((x-a)/(x+a))-1/(2*a^3)*atan(x/a)
+--R
+--R x - a x
+--R log(-----) - 2atan(-)
+--R x + a a
+--R (2) ---------------------
+--R 3
+--R 4a
+--R Type: Expression
Integer
+--E
+
+--S 41
+cc:=aa-bb
+--R
+--R x - a
+--R - log(x + a) + log(x - a) - log(-----)
+--R x + a
+--R (3) --------------------------------------
+--R 3
+--R 4a
+--R Type: Expression
Integer
+--E
+
+--S 42 14:318 Schaums and Axiom agree
+dd:=expandLog cc
+--R
+--R (4) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.319~~~~~$\displaystyle
@@ -277,7 +1019,7 @@ $$
<<*>>=
)clear all
---S 9 of 14
+--S 43
aa:=integrate(x/(x^4-a^4),x)
--R
--R
@@ -288,6 +1030,41 @@ aa:=integrate(x/(x^4-a^4),x)
--R 4a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 44
+bb:=1/(4*a^2)*log((x^2-a^2)/(x^2+a^2))
+--R
+--R 2 2
+--R x - a
+--R log(-------)
+--R 2 2
+--R x + a
+--R (2) ------------
+--R 2
+--R 4a
+--R Type: Expression
Integer
+--E
+
+--S 45
+cc:=aa-bb
+--R
+--R 2 2
+--R 2 2 2 2 x - a
+--R - log(x + a ) + log(x - a ) - log(-------)
+--R 2 2
+--R x + a
+--R (3) --------------------------------------------
+--R 2
+--R 4a
+--R Type: Expression
Integer
+--E
+
+--S 46 14:319 Schaums and Axiom agree
+dd:=expandLog cc
+--R
+--R (4) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.320~~~~~$\displaystyle
@@ -299,7 +1076,7 @@ $$
<<*>>=
)clear all
---S 10 of 14
+--S 47
aa:=integrate(x^2/(x^4-a^4),x)
--R
--R
@@ -309,7 +1086,36 @@ aa:=integrate(x^2/(x^4-a^4),x)
--R (1) ------------------------------------
--R 4a
--R Type: Union(Expression
Integer,...)
+--E
+
+--S 48
+bb:=1/(4*a)*log((x-a)/(x+a))+1/(2*a)*atan(x/a)
+--R
+--R x - a x
+--R log(-----) + 2atan(-)
+--R x + a a
+--R (2) ---------------------
+--R 4a
+--R Type: Expression
Integer
--E
+
+--S 49
+cc:=aa-bb
+--R
+--R x - a
+--R - log(x + a) + log(x - a) - log(-----)
+--R x + a
+--R (3) --------------------------------------
+--R 4a
+--R Type: Expression
Integer
+--E
+
+--S 50 14:320 Schaums and Axiom agree
+dd:=expandLog cc
+--R
+--R (4) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.321~~~~~$\displaystyle
@@ -320,7 +1126,7 @@ $$
<<*>>=
)clear all
---S 11 of 14
+--S 51
aa:=integrate(x^3/(x^4-a^4),x)
--R
--R
@@ -330,6 +1136,23 @@ aa:=integrate(x^3/(x^4-a^4),x)
--R 4
--R Type: Union(Expression
Integer,...)
--E
+
+--S 52
+bb:=1/4*log(x^4-a^4)
+--R
+--R 4 4
+--R log(x - a )
+--R (2) ------------
+--R 4
+--R Type: Expression
Integer
+--E
+
+--S 53 14:321 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.322~~~~~$\displaystyle
@@ -340,7 +1163,7 @@ $$
<<*>>=
)clear all
---S 12 of 14
+--S 54
aa:=integrate(1/(x*(x^4-a^4)),x)
--R
--R
@@ -350,7 +1173,42 @@ aa:=integrate(1/(x*(x^4-a^4)),x)
--R 4
--R 4a
--R Type: Union(Expression
Integer,...)
+--E
+
+--S 55
+bb:=1/(4*a^4)*log((x^4-a^4)/x^4)
+--R
+--R 4 4
+--R x - a
+--R log(-------)
+--R 4
+--R x
+--R (2) ------------
+--R 4
+--R 4a
+--R Type: Expression
Integer
--E
+
+--S 56
+cc:=aa-bb
+--R
+--R 4 4
+--R 4 4 x - a
+--R log(x - a ) - 4log(x) - log(-------)
+--R 4
+--R x
+--R (3) -------------------------------------
+--R 4
+--R 4a
+--R Type: Expression
Integer
+--E
+
+--S 57 14:322 Schaums and Axiom agree
+dd:=expandLog cc
+--R
+--R (4) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.323~~~~~$\displaystyle
@@ -362,7 +1220,7 @@ $$
<<*>>=
)clear all
---S 13 of 14
+--S 58
aa:=integrate(1/(x^2*(x^4-a^4)),x)
--R
--R
@@ -374,6 +1232,37 @@ aa:=integrate(1/(x^2*(x^4-a^4)),x)
--R 4a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 59
+bb:=1/(a^4*x)+1/(4*a^5)*log((x-a)/(x+a))+1/(2*a^5)*atan(x/a)
+--R
+--R x - a x
+--R x log(-----) + 2x atan(-) + 4a
+--R x + a a
+--R (2) ------------------------------
+--R 5
+--R 4a x
+--R Type: Expression
Integer
+--E
+
+--S 60
+cc:=aa-bb
+--R
+--R x - a
+--R - log(x + a) + log(x - a) - log(-----)
+--R x + a
+--R (3) --------------------------------------
+--R 5
+--R 4a
+--R Type: Expression
Integer
+--E
+
+--S 61 14:323 Schaums and Axiom agree
+dd:=expandLog cc
+--R
+--R (4) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.324~~~~~$\displaystyle
@@ -384,7 +1273,7 @@ $$
<<*>>=
)clear all
---S 14 of 14
+--S 62
aa:=integrate(1/(x^3*(x^4-a^4)),x)
--R
--R
@@ -396,6 +1285,40 @@ aa:=integrate(1/(x^3*(x^4-a^4)),x)
--R Type: Union(Expression
Integer,...)
--E
+--S 63
+bb:=1/(2*a^4*x^2)+1/(4*a^6)*log((x^2-a^2)/(x^2+a^2))
+--R
+--R 2 2
+--R 2 x - a 2
+--R x log(-------) + 2a
+--R 2 2
+--R x + a
+--R (2) --------------------
+--R 6 2
+--R 4a x
+--R Type: Expression
Integer
+--E
+
+--S 64
+cc:=aa-bb
+--R
+--R 2 2
+--R 2 2 2 2 x - a
+--R - log(x + a ) + log(x - a ) - log(-------)
+--R 2 2
+--R x + a
+--R (3) --------------------------------------------
+--R 6
+--R 4a
+--R Type: Expression
Integer
+--E
+
+--S 65 14:324 Schaums and Axiom agree
+dd:=expandLog cc
+--R
+--R (4) 0
+--R Type: Expression
Integer
+--E
)spool
)lisp (bye)
@
diff --git a/src/input/schaum16.input.pamphlet
b/src/input/schaum16.input.pamphlet
index f6b849c..b058a54 100644
--- a/src/input/schaum16.input.pamphlet
+++ b/src/input/schaum16.input.pamphlet
@@ -18,7 +18,7 @@ $$
)set message auto off
)clear all
---S 1 of 14
+--S 1
aa:=integrate(1/x*(x^n+a^n),x)
--R
--R
@@ -28,6 +28,79 @@ aa:=integrate(1/x*(x^n+a^n),x)
--R n
--R Type: Union(Expression
Integer,...)
--E
+
+--S 2
+bb:=1/(n*a^n)*log(x^n/(x^n+a^n))
+--R
+--R n
+--R x
+--R log(-------)
+--R n n
+--R x + a
+--R (2) ------------
+--R n
+--R n a
+--R Type: Expression
Integer
+--E
+
+--S 3
+cc:=aa-bb
+--R
+--R n
+--R x n n log(x) n 2
+--R - log(-------) + a %e + n log(x)(a )
+--R n n
+--R x + a
+--R (3) ---------------------------------------------
+--R n
+--R n a
+--R Type: Expression
Integer
+--E
+
+--S 4
+dd:=expandLog cc
+--R
+--R n n n n n log(x) 2n
+--R log(x + a ) - log(x ) + a %e + n log(x)a
+--R (4) ---------------------------------------------------
+--R n
+--R n a
+--R Type: Expression
Integer
+--E
+
+--S 5
+ee:=complexNormalize dd
+--R
+--R (5)
+--R n log(x) n log(a) n log(a) n log(x)
+--R log(%e + %e ) + %e %e
+--R +
+--R n log(a) 2
+--R n log(x)(%e ) - n log(x)
+--R /
+--R n log(a)
+--R n %e
+--R Type: Expression
Integer
+--E
+
+--S 6
+explog:=rule(%e^(n*log(x)) == x^n)
+--R
+--R n log(x) n
+--R (6) %e == x
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 7 14:325 Axiom cannot simplify this expression
+ff:=explog ee
+--R
+--R n n n n 2n
+--R log(x + a ) + a x + n log(x)a - n log(x)
+--R (7) --------------------------------------------
+--R n
+--R n a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.326~~~~~$\displaystyle
@@ -38,7 +111,7 @@ $$
<<*>>=
)clear all
---S 2 of 14
+--S 8
aa:=integrate(x^(n-1)/(x^n+a^n),x)
--R
--R
@@ -48,6 +121,41 @@ aa:=integrate(x^(n-1)/(x^n+a^n),x)
--R n
--R Type: Union(Expression
Integer,...)
--E
+
+--S 9
+bb:=1/n*log(x^n+a^n)
+--R
+--R n n
+--R log(x + a )
+--R (2) ------------
+--R n
+--R Type: Expression
Integer
+--E
+
+--S 10
+cc:=aa-bb
+--R
+--R n log(x) n n n
+--R log(%e + a ) - log(x + a )
+--R (3) -----------------------------------
+--R n
+--R Type: Expression
Integer
+--E
+
+--S 11
+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 12 14:326 Schaums and Axiom agree
+dd:=explog cc
+--R
+--R (5) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.327~~~~~$\displaystyle
@@ -59,7 +167,7 @@ $$
<<*>>=
)clear all
---S 3 of 14
+--S 13 14:327 Axiom cannot compute this integral
aa:=integrate(x^m/(x^n+a^n)^r,x)
--R
--R
@@ -81,7 +189,7 @@ $$
<<*>>=
)clear all
---S 4 of 14
+--S 14 14:328 Axiom cannot compute this integral
aa:=integrate(1/(x^m*(x^n+a^n)^r),x)
--R
--R
@@ -103,7 +211,7 @@ $$
<<*>>=
)clear all
---S 5 of 14
+--S 15
aa:=integrate(1/(x*sqrt(x^n+a^n)),x)
--R
--R
@@ -130,6 +238,114 @@ aa:=integrate(1/(x*sqrt(x^n+a^n)),x)
--R n\|- a
--R Type: Union(List Expression
Integer,...)
--E
+
+--S 16
+bb:=1/(n*sqrt(a^n))*log((sqrt(x^n+a^n)-sqrt(a^n))/(sqrt(x^n+a^n)+sqrt(a^n)))
+--R
+--R +-------+ +--+
+--R | n n | n
+--R \|x + a - \|a
+--R log(------------------)
+--R +-------+ +--+
+--R | n n | n
+--R \|x + a + \|a
+--R (2) -----------------------
+--R +--+
+--R | n
+--R n\|a
+--R Type: Expression
Integer
+--E
+
+--S 17
+cc1:=aa.1-bb
+--R
+--R (3)
+--R +---------------+ +--+
+--R n | n log(x) n n log(x) n | n
+--R - 2a \|%e + a + (%e + 2a )\|a
+--R log(-------------------------------------------------)
+--R n log(x)
+--R %e
+--R +
+--R +-------+ +--+
+--R | n n | n
+--R \|x + a - \|a
+--R - log(------------------)
+--R +-------+ +--+
+--R | n n | n
+--R \|x + a + \|a
+--R /
+--R +--+
+--R | n
+--R n\|a
+--R Type: Expression
Integer
+--E
+
+--S 18
+dd1:=expandLog cc1
+--R
+--R (4)
+--R +---------------+ +--+
+--R n | n log(x) n n log(x) n | n
+--R log(2a \|%e + a + (- %e - 2a )\|a )
+--R +
+--R +-------+ +--+ +-------+ +--+
+--R | n n | n | n n | n
+--R log(\|x + a + \|a ) - log(\|x + a - \|a ) - n log(x) + log(-
1)
+--R /
+--R +--+
+--R | n
+--R n\|a
+--R Type: Expression
Integer
+--E
+
+--S 19
+explog:=rule(%e^(n*log(x)) == x^n)
+--R
+--R n log(x) n
+--R (5) %e == x
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 20
+ee1:=explog dd1
+--R
+--R (6)
+--R +-------+ +--+ +-------+ +--+
+--R n | n n n n | n | n n | n
+--R log(2a \|x + a + (- x - 2a )\|a ) + log(\|x + a + \|a )
+--R +
+--R +-------+ +--+
+--R | n n | n
+--R - log(\|x + a - \|a ) - n log(x) + log(- 1)
+--R /
+--R +--+
+--R | n
+--R n\|a
+--R Type: Expression
Integer
+--E
+
+--S 21
+ff1:=complexNormalize ee1
+--R
+--R n log(a) + 4log(- 1)
+--R (7) --------------------
+--R +----------+
+--R | n log(a)
+--R 2n\|%e
+--R Type: Expression
Integer
+--E
+
+--S 22 14:329 Schaums and Axiom differ by a constant
+gg1:=explog ff1
+--R
+--R n log(a) + 4log(- 1)
+--R (8) --------------------
+--R +--+
+--R | n
+--R 2n\|a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.330~~~~~$\displaystyle
@@ -140,7 +356,7 @@ $$
<<*>>=
)clear all
---S 6 of 14
+--S 23
aa:=integrate(1/(x*(x^n-a^n)),x)
--R
--R
@@ -151,6 +367,79 @@ aa:=integrate(1/(x*(x^n-a^n)),x)
--R n a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 24
+bb:=1/(n*a^n)*log((x^n-a^n)/x^n)
+--R
+--R n n
+--R x - a
+--R log(-------)
+--R n
+--R x
+--R (2) ------------
+--R n
+--R n a
+--R Type: Expression
Integer
+--E
+
+--S 25
+cc:=aa-bb
+--R
+--R n n
+--R n log(x) n x - a
+--R log(%e - a ) - log(-------) - n log(x)
+--R n
+--R x
+--R (3) ----------------------------------------------
+--R n
+--R n a
+--R Type: Expression
Integer
+--E
+
+--S 26
+dd:=expandLog cc
+--R
+--R n log(x) n n n n
+--R log(%e - a ) + log(x ) - log(x - a ) - n log(x)
+--R (4) --------------------------------------------------------
+--R n
+--R n a
+--R Type: Expression
Integer
+--E
+
+--S 27
+explog:=rule(%e^(n*log(x)) == x^n)
+--R
+--R n log(x) n
+--R (5) %e == x
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 28
+ee:=explog dd
+--R
+--R n
+--R log(x ) - n log(x)
+--R (6) ------------------
+--R n
+--R n a
+--R Type: Expression
Integer
+--E
+
+--S 29
+logpow:=rule(log(a^n) == n*log(a))
+--R
+--R n
+--R (7) log(a ) == n log(a)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 30 14:330 Schaums and Axiom agree
+ff:=logpow ee
+--R
+--R (8) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.331~~~~~$\displaystyle
@@ -161,7 +450,7 @@ $$
<<*>>=
)clear all
---S 7 of 14
+--S 31
aa:=integrate(x^(n-1)/(x^n-a^n),x)
--R
--R
@@ -171,6 +460,42 @@ aa:=integrate(x^(n-1)/(x^n-a^n),x)
--R n
--R Type: Union(Expression
Integer,...)
--E
+
+--S 32
+bb:=1/n*log(x^n-a^n)
+--R
+--R n n
+--R log(x - a )
+--R (2) ------------
+--R n
+--R Type: Expression
Integer
+--E
+
+--S 33
+cc:=aa-bb
+--R
+--R n log(x) n n n
+--R log(%e - a ) - log(x - a )
+--R (3) -----------------------------------
+--R n
+--R Type: Expression
Integer
+--E
+
+--S 34
+explog:=rule(%e^(n*log(x)) == x^n)
+--R
+--R n log(x) n
+--R (4) %e == x
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 35 14:331 Schaums and Axiom agree
+dd:=explog cc
+--R
+--R (5) 0
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.332~~~~~$\displaystyle
@@ -182,7 +507,7 @@ $$
<<*>>=
)clear all
---S 8 of 14
+--S 36 14:332 Axiom cannot compute this integral
aa:=integrate(x^m/(x^n-a^n)^r,x)
--R
--R
@@ -204,7 +529,7 @@ $$
<<*>>=
)clear all
---S 9 of 14
+--S 37 14:333 Axiom cannot compute this integral
aa:=integrate(1/(x^m*(x^n-a^n)^r),x)
--R
--R
@@ -225,7 +550,7 @@ $$
<<*>>=
)clear all
---S 10 of 14
+--S 38
aa:=integrate(1/(x*sqrt(x^n-a^n)),x)
--R
--R
@@ -252,6 +577,62 @@ aa:=integrate(1/(x*sqrt(x^n-a^n)),x)
--R n\|a
--R Type: Union(List Expression
Integer,...)
--E
+
+--S 39
+bb:=2/(n*sqrt(a^n))*acos(sqrt(a^n/x^n))
+--R
+--R +--+
+--R | n
+--R |a
+--R 2acos( |-- )
+--R | n
+--R \|x
+--R (2) ------------
+--R +--+
+--R | n
+--R n\|a
+--R Type: Expression
Integer
+--E
+
+--S 40
+cc1:=aa.1-bb
+--R
+--R (3)
+--R +---------------+ +----+
+--R +--+ n | n log(x) n n log(x) n | n
+--R | n 2a \|%e - a + (%e - 2a )\|- a
+--R \|a log(-------------------------------------------------)
+--R n log(x)
+--R %e
+--R +
+--R +--+
+--R +----+ | n
+--R | n |a
+--R - 2\|- a acos( |-- )
+--R | n
+--R \|x
+--R /
+--R +----+ +--+
+--R | n | n
+--R n\|- a \|a
+--R Type: Expression
Integer
+--E
+
+--S 41 14:334 Axiom cannot simplify this expression
+cc2:=aa.2-bb
+--R
+--R +--+ +---------------+ +--+
+--R | n | n log(x) n | n
+--R \|a \|%e - a |a
+--R 2atan(-----------------------) - 2acos( |-- )
+--R n | n
+--R a \|x
+--R (4) ---------------------------------------------
+--R +--+
+--R | n
+--R n\|a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.335~~~~~$\displaystyle
@@ -269,7 +650,7 @@ $$
<<*>>=
)clear all
---S 11 of 14
+--S 42 14:335 Axiom cannot compute this integral
aa:=integrate(x^(p-1)/(x^(2*m)+a^(2*m)),x)
--R
--R
@@ -299,7 +680,7 @@ $$
<<*>>=
)clear all
---S 12 of 14
+--S 43 14:336 Axiom cannot compute this integral
aa:=integrate(x^(p-1)/(x^(2*m)-a^(2*m)),x)
--R
--R
@@ -333,7 +714,7 @@ $$
<<*>>=
)clear all
---S 13 of 14
+--S 44 14:337 Axiom cannot compute this integral
aa:=integrate(x^(p-1)/(x^(2*m+1)+a^(2*m+1)),x)
--R
--R
@@ -369,7 +750,7 @@ $$
<<*>>=
)clear all
---S 14 of 14
+--S 45 14:338 Axiom cannot compute this integral
aa:=integrate(x^(p-1)/(x^(2*m+1)-a^(2*m+1)),x)
--R
--R
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- [Axiom-developer] 20080421.01.tpd.patch (CATS Schaum2-Axiom equivalence testing (14-16)),
daly <=