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[Axiom-developer] 20080424.01.tpd.patch (CATS Shaums-Axiom equivalence t
|
From: |
daly |
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Subject: |
[Axiom-developer] 20080424.01.tpd.patch (CATS Shaums-Axiom equivalence testing (23)) |
|
Date: |
Fri, 25 Apr 2008 01:24:24 -0500 |
schaum23.input.pamphlet
14:461 Schaums and Axiom agree
14:462 Schaums and Axiom agree
14:463 Schaums and Axiom agree
14:464 Axiom cannot simplify this expression
14:465 Schaums and Axiom agree
14:466 Axiom cannot compute this integral
14:467 Axiom cannot compute this integral
14:468 Schaums and Axiom differ by a constant
14:469 Schaums and Axiom agree
14:470 Axiom cannot compute this integral
======================================================================
diff --git a/changelog b/changelog
index 80e94f3..314b281 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,4 @@
+20080424 tpd src/input/schaum23.input show Schaums-Axiom equivalence
20080423 tpd src/input/schaum22.input show Schaums-Axiom equivalence
20080423 tpd src/input/schaum21.input show Schaums-Axiom equivalence
20080423 tpd src/input/schaum20.input show Schaums-Axiom equivalence
diff --git a/src/input/schaum23.input.pamphlet
b/src/input/schaum23.input.pamphlet
index 679e775..c24ef73 100644
--- a/src/input/schaum23.input.pamphlet
+++ b/src/input/schaum23.input.pamphlet
@@ -19,7 +19,7 @@ $$
)set message auto off
)clear all
---S 1 of 10
+--S 1
aa:=integrate(csc(a*x),x)
--R
--R
@@ -30,6 +30,148 @@ aa:=integrate(csc(a*x),x)
--R a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 2
+bb1:=1/a*log(csc(a*x)-cot(a*x))
+--R
+--R log(csc(a x) - cot(a x))
+--R (2) ------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 3
+bb2:=1/a*log(tan((a*x)/2))
+--R
+--R a x
+--R log(tan(---))
+--R 2
+--R (3) -------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 4
+cc1:=aa-bb1
+--R
+--R sin(a x)
+--R log(------------) - log(csc(a x) - cot(a x))
+--R cos(a x) + 1
+--R (4) --------------------------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 5
+cotrule:=rule(cot(a) == cos(a)/sin(a))
+--R
+--R cos(a)
+--R (5) cot(a) == ------
+--R sin(a)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 6
+dd1:=cotrule cc1
+--R
+--R sin(a x) csc(a x)sin(a x) - cos(a x)
+--R log(------------) - log(---------------------------)
+--R cos(a x) + 1 sin(a x)
+--R (6) ----------------------------------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 7
+cscrule:=rule(csc(a) == 1/sin(a))
+--R
+--R 1
+--R (7) csc(a) == ------
+--R sin(a)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 8
+ee1:=cscrule dd1
+--R
+--R sin(a x) - cos(a x) + 1
+--R log(------------) - log(--------------)
+--R cos(a x) + 1 sin(a x)
+--R (8) ---------------------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 9
+ff1:=expandLog ee1
+--R
+--R 2log(sin(a x)) - log(cos(a x) + 1) - log(cos(a x) - 1) - log(- 1)
+--R (9) -----------------------------------------------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 10
+gg1:=complexNormalize ff1
+--R
+--R 2log(- 1)
+--R (10) - ---------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 11
+cc2:=aa-bb2
+--R
+--R a x sin(a x)
+--R - log(tan(---)) + log(------------)
+--R 2 cos(a x) + 1
+--R (11) -----------------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 12
+tanrule:=rule(tan(a) == sin(a)/cos(a))
+--R
+--R sin(a)
+--R (12) tan(a) == ------
+--R cos(a)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 13
+dd2:=tanrule cc2
+--R
+--R a x
+--R sin(---)
+--R sin(a x) 2
+--R log(------------) - log(--------)
+--R cos(a x) + 1 a x
+--R cos(---)
+--R 2
+--R (13) ---------------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 14
+ee2:=expandLog dd2
+--R
+--R a x a x
+--R log(sin(a x)) - log(sin(---)) - log(cos(a x) + 1) + log(cos(---))
+--R 2 2
+--R (14) -----------------------------------------------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 15 14:461 Schaums and Axiom agree
+ff2:=complexNormalize ee2
+--R
+--R (15) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.462~~~~~$\displaystyle
@@ -40,7 +182,7 @@ $$
<<*>>=
)clear all
---S 2 of 10
+--S 16
aa:=integrate(csc(a*x)^2,x)
--R
--R
@@ -49,6 +191,40 @@ aa:=integrate(csc(a*x)^2,x)
--R a sin(a x)
--R Type: Union(Expression
Integer,...)
--E
+
+--S 17
+bb:=-cot(a*x)/a
+--R
+--R cot(a x)
+--R (2) - --------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 18
+cc:=aa-bb
+--R
+--R cot(a x)sin(a x) - cos(a x)
+--R (3) ---------------------------
+--R a sin(a x)
+--R Type: Expression
Integer
+--E
+
+--S 19
+cotrule:=rule(cot(a) == cos(a)/sin(a))
+--R
+--R cos(a)
+--R (4) cot(a) == ------
+--R sin(a)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 20 14:462 Schaums and Axiom agree
+dd:=cotrule cc
+--R
+--R (5) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.463~~~~~$\displaystyle
@@ -59,7 +235,7 @@ $$
<<*>>=
)clear all
---S 3 of 10
+--S 21
aa:=integrate(csc(a*x)^3,x)
--R
--R
@@ -71,6 +247,159 @@ aa:=integrate(csc(a*x)^3,x)
--R 2a cos(a x) - 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 22
+bb:=-(csc(a*x)*cot(a*x))/(2*a)+1/(2*a)*log(tan((a*x)/2))
+--R
+--R a x
+--R log(tan(---)) - cot(a x)csc(a x)
+--R 2
+--R (2) --------------------------------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 23
+cc:=aa-bb
+--R
+--R (3)
+--R 2 a x 2 sin(a x)
+--R (- cos(a x) + 1)log(tan(---)) + (cos(a x) - 1)log(------------)
+--R 2 cos(a x) + 1
+--R +
+--R 2
+--R (cos(a x) - 1)cot(a x)csc(a x) + cos(a x)
+--R /
+--R 2
+--R 2a cos(a x) - 2a
+--R Type: Expression
Integer
+--E
+
+--S 24
+cotrule:=rule(cot(a) == cos(a)/sin(a))
+--R
+--R cos(a)
+--R (4) cot(a) == ------
+--R sin(a)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 25
+dd:=cotrule cc
+--R
+--R (5)
+--R 2 a x
+--R (- cos(a x) + 1)sin(a x)log(tan(---))
+--R 2
+--R +
+--R 2 sin(a x)
+--R (cos(a x) - 1)sin(a x)log(------------) + cos(a x)sin(a x)
+--R cos(a x) + 1
+--R +
+--R 3
+--R (cos(a x) - cos(a x))csc(a x)
+--R /
+--R 2
+--R (2a cos(a x) - 2a)sin(a x)
+--R Type: Expression
Integer
+--E
+
+--S 26
+tanrule:=rule(tan(a) == sin(a)/cos(a))
+--R
+--R sin(a)
+--R (6) tan(a) == ------
+--R cos(a)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 27
+ee:=tanrule dd
+--R
+--R (7)
+--R 2 sin(a x)
+--R (cos(a x) - 1)sin(a x)log(------------)
+--R cos(a x) + 1
+--R +
+--R a x
+--R sin(---)
+--R 2 2
+--R (- cos(a x) + 1)sin(a x)log(--------) + cos(a x)sin(a x)
+--R a x
+--R cos(---)
+--R 2
+--R +
+--R 3
+--R (cos(a x) - cos(a x))csc(a x)
+--R /
+--R 2
+--R (2a cos(a x) - 2a)sin(a x)
+--R Type: Expression
Integer
+--E
+
+--S 28
+cscrule:=rule(csc(a) == 1/sin(a))
+--R
+--R 1
+--R (8) csc(a) == ------
+--R sin(a)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 29
+ff:=cscrule ee
+--R
+--R (9)
+--R 2 2 sin(a x)
+--R (cos(a x) - 1)sin(a x) log(------------)
+--R cos(a x) + 1
+--R +
+--R a x
+--R sin(---)
+--R 2 2 2 2
3
+--R (- cos(a x) + 1)sin(a x) log(--------) + cos(a x)sin(a x) + cos(a
x)
+--R a x
+--R cos(---)
+--R 2
+--R +
+--R - cos(a x)
+--R /
+--R 2 2
+--R (2a cos(a x) - 2a)sin(a x)
+--R Type: Expression
Integer
+--E
+
+--S 30
+gg:=expandLog ff
+--R
+--R (10)
+--R 2 2
+--R (cos(a x) - 1)sin(a x) log(sin(a x))
+--R +
+--R 2 2 a x
+--R (- cos(a x) + 1)sin(a x) log(sin(---))
+--R 2
+--R +
+--R 2 2
+--R (- cos(a x) + 1)sin(a x) log(cos(a x) + 1)
+--R +
+--R 2 2 a x 2 3
+--R (cos(a x) - 1)sin(a x) log(cos(---)) + cos(a x)sin(a x) + cos(a x)
+--R 2
+--R +
+--R - cos(a x)
+--R /
+--R 2 2
+--R (2a cos(a x) - 2a)sin(a x)
+--R Type: Expression
Integer
+--E
+
+--S 31 14:463 Schaums and Axiom agree
+hh:=complexNormalize gg
+--R
+--R (11) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.464~~~~~$\displaystyle
@@ -81,7 +410,7 @@ $$
<<*>>=
)clear all
---S 4 of 10
+--S 32
aa:=integrate(csc(a*x)^n*cot(a*x),x)
--R
--R
@@ -96,6 +425,31 @@ aa:=integrate(csc(a*x)^n*cot(a*x),x)
--R a n
--R Type: Union(Expression
Integer,...)
--E
+
+--S 33
+bb:=-csc(a*x)^n/(n*a)
+--R
+--R n
+--R csc(a x)
+--R (2) - ---------
+--R a n
+--R Type: Expression
Integer
+--E
+
+--S 34 14:464 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R 1
+--R n log(- -------------)
+--R 2
+--R cos(a x) - 1
+--R ----------------------
+--R 2 n
+--R - %e + csc(a x)
+--R (3) --------------------------------------
+--R a n
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.465~~~~~$\displaystyle
@@ -106,7 +460,7 @@ $$
<<*>>=
)clear all
---S 5 of 10
+--S 35
aa:=integrate(1/csc(a*x),x)
--R
--R
@@ -114,7 +468,23 @@ aa:=integrate(1/csc(a*x),x)
--R (1) - --------
--R a
--R Type: Union(Expression
Integer,...)
+--E
+
+--S 36
+bb:=-cos(a*x)/a
+--R
+--R cos(a x)
+--R (2) - --------
+--R a
+--R Type: Expression
Integer
--E
+
+--S 37 14:465 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.466~~~~~$\displaystyle
@@ -126,7 +496,7 @@ $$
<<*>>=
)clear all
---S 6 of 10
+--S 38 14:466 Axiom cannot compute this integral
aa:=integrate(x*csc(a*x),x)
--R
--R
@@ -147,7 +517,7 @@ $$
<<*>>=
)clear all
---S 7 of 10
+--S 39 14:467 Axiom cannot compute this integral
aa:=integrate(csc(a*x)/x,x)
--R
--R
@@ -167,7 +537,7 @@ $$
<<*>>=
)clear all
---S 8 of 10
+--S 40
aa:=integrate(x*csc(a*x)^2,x)
--R
--R
@@ -179,6 +549,64 @@ aa:=integrate(x*csc(a*x)^2,x)
--R a sin(a x)
--R Type: Union(Expression
Integer,...)
--E
+
+--S 41
+bb:=-(x*cot(a*x))/a+1/a^2*log(sin(a*x))
+--R
+--R log(sin(a x)) - a x cot(a x)
+--R (2) ----------------------------
+--R 2
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 42
+cc:=aa-bb
+--R
+--R (3)
+--R sin(a x)
+--R - sin(a x)log(sin(a x)) + sin(a x)log(------------)
+--R cos(a x) + 1
+--R +
+--R 2
+--R - sin(a x)log(------------) + a x cot(a x)sin(a x) - a x cos(a x)
+--R cos(a x) + 1
+--R /
+--R 2
+--R a sin(a x)
+--R Type: Expression
Integer
+--E
+
+--S 43
+cotrule:=rule(cot(a) == cos(a)/sin(a))
+--R
+--R cos(a)
+--R (4) cot(a) == ------
+--R sin(a)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 44
+dd:=cotrule cc
+--R
+--R sin(a x) 2
+--R - log(sin(a x)) + log(------------) - log(------------)
+--R cos(a x) + 1 cos(a x) + 1
+--R (5) -------------------------------------------------------
+--R 2
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 45 14:468 Schaums and Axiom differ by a constant
+ee:=expandLog dd
+--R
+--R log(2)
+--R (6) - ------
+--R 2
+--R a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.469~~~~~$\displaystyle
@@ -189,7 +617,7 @@ $$
<<*>>=
)clear all
---S 9 of 10
+--S 46
aa:=integrate(1/(q+p*csc(a*x)),x)
--R
--R
@@ -227,6 +655,156 @@ aa:=integrate(1/(q+p*csc(a*x)),x)
--R a q\|- q + p
--R Type: Union(List Expression
Integer,...)
--E
+
+--S 47
+t1:=integrate(1/(p+q*sin(a*x)),x)
+--E
+
+--S 48
+bb1:=x/q-p/q*t1.1
+--R
+--R (3)
+--R -
+--R p
+--R *
+--R log
+--R +-------+
+--R 2 2 2 | 2 2
+--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p
+--R +
+--R 2 3 3 2 3 2
+--R (- p q + p )sin(a x) + (- q + p q)cos(a x) - q + p q
+--R /
+--R q sin(a x) + p
+--R +
+--R +-------+
+--R | 2 2
+--R a x\|q - p
+--R /
+--R +-------+
+--R | 2 2
+--R a q\|q - p
+--R Type: Expression
Integer
+--E
+
+--S 49
+bb2:=x/q-p/q*t1.2
+--R
+--R +---------+
+--R | 2 2 +---------+
+--R (p sin(a x) + q cos(a x) + q)\|- q + p | 2 2
+--R 2p atan(-----------------------------------------) + a x\|- q + p
+--R 2 2 2 2
+--R (q - p )cos(a x) + q - p
+--R (4) --------------------------------------------------------------------
+--R +---------+
+--R | 2 2
+--R a q\|- q + p
+--R Type: Expression
Integer
+--E
+
+--S 50
+cc1:=aa.1-bb1
+--R
+--R (5)
+--R p
+--R *
+--R log
+--R +-------+
+--R 2 2 2 | 2 2
+--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p
+--R +
+--R 2 3 3 2 3 2
+--R (p q - p )sin(a x) + (q - p q)cos(a x) + q - p q
+--R /
+--R q sin(a x) + p
+--R +
+--R p
+--R *
+--R log
+--R +-------+
+--R 2 2 2 | 2 2
+--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p
+--R +
+--R 2 3 3 2 3 2
+--R (- p q + p )sin(a x) + (- q + p q)cos(a x) - q + p q
+--R /
+--R q sin(a x) + p
+--R /
+--R +-------+
+--R | 2 2
+--R a q\|q - p
+--R Type: Expression
Integer
+--E
+
+--S 51
+cc2:=aa.2-bb1
+--R
+--R (6)
+--R +---------+
+--R | 2 2
+--R p\|- q + p
+--R *
+--R log
+--R +-------+
+--R 2 2 2 | 2 2
+--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p
+--R +
+--R 2 3 3 2 3 2
+--R (- p q + p )sin(a x) + (- q + p q)cos(a x) - q + p q
+--R /
+--R q sin(a x) + p
+--R +
+--R +---------+
+--R +-------+ | 2 2
+--R | 2 2 (p sin(a x) + q cos(a x) + q)\|- q + p
+--R 2p\|q - p atan(-----------------------------------------)
+--R 2 2 2 2
+--R (q - p )cos(a x) + q - p
+--R /
+--R +---------+ +-------+
+--R | 2 2 | 2 2
+--R a q\|- q + p \|q - p
+--R Type: Expression
Integer
+--E
+
+--S 52
+cc3:=aa.1-bb2
+--R
+--R (7)
+--R +---------+
+--R | 2 2
+--R p\|- q + p
+--R *
+--R log
+--R +-------+
+--R 2 2 2 | 2 2
+--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p
+--R +
+--R 2 3 3 2 3 2
+--R (p q - p )sin(a x) + (q - p q)cos(a x) + q - p q
+--R /
+--R q sin(a x) + p
+--R +
+--R +---------+
+--R +-------+ | 2 2
+--R | 2 2 (p sin(a x) + q cos(a x) + q)\|- q + p
+--R - 2p\|q - p atan(-----------------------------------------)
+--R 2 2 2 2
+--R (q - p )cos(a x) + q - p
+--R /
+--R +---------+ +-------+
+--R | 2 2 | 2 2
+--R a q\|- q + p \|q - p
+--R Type: Expression
Integer
+--E
+
+--S 53 14:469 Schaums and Axiom agree
+cc4:=aa.2-bb2
+--R
+--R (8) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.470~~~~~$\displaystyle
@@ -238,7 +816,7 @@ $$
<<*>>=
)clear all
---S 10 of 10
+--S 54 14:470 Axiom cannot compute this integral
aa:=integrate(csc(a*x)^n,x)
--R
--R
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- [Axiom-developer] 20080424.01.tpd.patch (CATS Shaums-Axiom equivalence testing (23)),
daly <=