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## Re: [Axiom-developer] 20080416.01.tpd.patch (CATS Schaums-Axiom equivale

 From: Doug Stewart Subject: Re: [Axiom-developer] 20080416.01.tpd.patch (CATS Schaums-Axiom equivalence testing (2-7)) Date: Thu, 17 Apr 2008 06:26:26 -0400 User-agent: Thunderbird 2.0.0.12 (X11/20080227)

address@hidden wrote:

Item   14:150 Schaums and Axiom DISAGREE BY A NON-CONSTANT
in schaum7.input.pamphlet is particularly interesting because
it appears that the derivative of Axiom's answer is the original
integrand but the derivative of Schaum's answer is not, implying

that Schaum's has a mistake. This will be verified using other systems later.

Axiom is weak in handling certain simplifications. Future work is
planned to correct this.

Richard Fateman has given me permission to use his TILU pattern
integration database in Axiom. This should give us much broader
integration results. TILU has not been tested against Schaums
but this testing will occur during the merge.

schaum2.input.pamphet

14:150 Schaums and Axiom DISAGREE BY A NON-CONSTANT

\section{\cite{1}:14.150~~~~~$\displaystyle\int{\frac{dx}{x^3(x^2-a^2)}}$}
@@ -124,7 +343,7 @@ 
<<*>>=
)clear all

---S 7 of 19
+--S 29
aa:=integrate(1/(x^3*(x^2-a^2)),x)

--R --R
@@ -135,6 +354,73 @@ aa:=integrate(1/(x^3*(x^2-a^2)),x)
--R                     2a x
--R                                          Type: Union(Expression
Integer,...)

--E +
+--S 30
+bb:=1/(2*a^2*x*2)-1/(2*a^4)*log(x^2/(x^2-a^2))
+--R
+--R                     2
+--R                    x        2
+--R        - 2x log(-------) + a
+--R                  2    2
+--R                 x  - a
+--R   (2)  ----------------------
+--R                   4
+--R                 4a x
+--R                                                     Type: Expression
Integer
+--E
+
+--S 31
+cc:=aa-bb
+--R
+--R                                                 2
+--R          2     2    2      2           2       x        2      2
+--R        2x log(x  - a ) - 4x log(x) + 2x log(-------) - a x + 2a
+--R                                              2    2
+--R                                             x  - a
+--R   (3)  ---------------------------------------------------------
+--R                                    4 2
+--R                                  4a x
+--R                                                     Type: Expression
Integer
+--E
+
+--S 32
+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 33
+dd:=divlog cc
+--R
+--R          2     2      2          2      2
+--R        2x log(x ) - 4x log(x) - a x + 2a
+--R   (5)  ----------------------------------
+--R                         4 2
+--R                       4a x
+--R                                                     Type: Expression
Integer
+--E
+
+--S 34
+logpow:=rule(log(a^n) == n*log(a))
+--R
+--R             n
+--R   (6)  log(a ) == n log(a)
+--R                        Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 35     14:150 Schaums and Axiom DISAGREE BY A NON-CONSTANT
+ee:=logpow dd
+--R
+--R        - x + 2
+--R   (7)  -------
+--R           2 2
+--R         4a x
+--R                                                     Type: Expression
Integer
+--E
+
@



I tried it on maximum and got:

(%i6) integrate(1/(x^3*(x^2-a^2)),x);
2      2
log(x  - a )            log(x)                    1

(%o6) ------------ - ------ + ------- 4 4 2 2 2 a a 2 a x

and this agrees with my schaums book;