MathAction #290 Axiom failed to integrate elementary function

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When I type:

fricas

(1) -> ex1:=(x-x/(x+exp(x)/(x*log(x+x))))*exp(2)

$\label{eq1}\frac{{{\left({{x}^{3}}-{{x}^{2}}\right)}\ {{e}^{2}}\ {\log \left({2 \ x}\right)}}+{x \ {{e}^{2}}\ {{e}^{x}}}}{{{{x}^{2}}\ {\log \left({2 \ x}\right)}}+{{e}^{x}}}$

(1)

Type: Expression(Integer)

fricas

ex2:=D(ex1, x)

$\label{eq2}\frac{{{{x}^{4}}\ {{e}^{2}}\ {{\log \left({2 \ x}\right)}^{2}}}+{{\left({3 \ {{x}^{2}}}-{2 \ x}\right)}\ {{e}^{2}}\ {{e}^{x}}\ {\log \left({2 \ x}\right)}}+{{{e}^{2}}\ {{{e}^{x}}^{2}}}-{x \ {{e}^{2}}\ {{e}^{x}}}}{{{{x}^{4}}\ {{\log \left({2 \ x}\right)}^{2}}}+{2 \ {{x}^{2}}\ {{e}^{x}}\ {\log \left({2 \ x}\right)}}+{{{e}^{x}}^{2}}}$

(2)

Type: Expression(Integer)

fricas

ex3:=simplify(ex2)

$\label{eq3}\frac{{{{x}^{4}}\ {{e}^{2}}\ {{\log \left({2 \ x}\right)}^{2}}}+{{\left({3 \ {{x}^{2}}}-{2 \ x}\right)}\ {{e}^{x + 2}}\ {\log \left({2 \ x}\right)}}+{{e}^{{2 \ x}+ 2}}-{x \ {{e}^{x + 2}}}}{{{{x}^{4}}\ {{\log \left({2 \ x}\right)}^{2}}}+{2 \ {{x}^{2}}\ {{e}^{x}}\ {\log \left({2 \ x}\right)}}+{{e}^{2 \ x}}}$

(3)

Type: Expression(Integer)

fricas

integrate(ex3, x)

$\label{eq4}\frac{{{{x}^{3}}\ {{{e}^{2}}^{2}}\ {\log \left({2 \ x}\right)}}+{{\left(x + 1 \right)}\ {{e}^{2}}\ {{e}^{x + 2}}}}{{{{x}^{2}}\ {{e}^{2}}\ {\log \left({2 \ x}\right)}}+{{e}^{x + 2}}}$

(4)

Type: Union(Expression(Integer),...)

then Axiom 3.9 returns unevaluated integral.

Note, that if I remove the exp(2) factor, then Axiom is able to compute the integral.

fricas

ex4:=(x-x/(x+exp(x)/(x*log(x+x))))

$\label{eq5}\frac{{{\left({{x}^{3}}-{{x}^{2}}\right)}\ {\log \left({2 \ x}\right)}}+{x \ {{e}^{x}}}}{{{{x}^{2}}\ {\log \left({2 \ x}\right)}}+{{e}^{x}}}$

(5)

Type: Expression(Integer)

fricas

ex5:=D(ex4, x)

$\label{eq6}\frac{{{{x}^{4}}\ {{\log \left({2 \ x}\right)}^{2}}}+{{\left({3 \ {{x}^{2}}}-{2 \ x}\right)}\ {{e}^{x}}\ {\log \left({2 \ x}\right)}}+{{{e}^{x}}^{2}}-{x \ {{e}^{x}}}}{{{{x}^{4}}\ {{\log \left({2 \ x}\right)}^{2}}}+{2 \ {{x}^{2}}\ {{e}^{x}}\ {\log \left({2 \ x}\right)}}+{{{e}^{x}}^{2}}}$

(6)

Type: Expression(Integer)

fricas

ex6:=simplify(ex5)

$\label{eq7}\frac{{{{x}^{4}}\ {{\log \left({2 \ x}\right)}^{2}}}+{{\left({3 \ {{x}^{2}}}-{2 \ x}\right)}\ {{e}^{x}}\ {\log \left({2 \ x}\right)}}+{{e}^{2 \ x}}-{x \ {{e}^{x}}}}{{{{x}^{4}}\ {{\log \left({2 \ x}\right)}^{2}}}+{2 \ {{x}^{2}}\ {{e}^{x}}\ {\log \left({2 \ x}\right)}}+{{e}^{2 \ x}}}$

(7)

Type: Expression(Integer)

fricas

integrate(ex6, x)

$\label{eq8}\frac{{{{x}^{3}}\ {\log \left({2 \ x}\right)}}+{{\left(x + 1 \right)}\ {{e}^{x}}}}{{{{x}^{2}}\ {\log \left({2 \ x}\right)}}+{{e}^{x}}}$

(8)

Type: Union(Expression(Integer),...)

We claim that Axiom is good at integration, therefore critical --kratt6, Wed, 10 May 2006 01:28:56 -0500 reply

Severity: normal => critical

We claim that Axiom is good at integration, therefore critical --kratt6, Wed, 10 May 2006 01:29:14 -0500 reply

Category: Axiom Mathematics => Axiom Library

Related example --unknown, Wed, 10 May 2006 06:22:25 -0500 reply

If I try:

fricas

integrate(simplify(((x^4*log(2*x)^2+(3*x^2-2*x)*exp(x)*log(2*x)+exp(2*x)-x*exp(x))/(x^4*log(2*x)^2+2*x^2*exp(x)*log(2*x)+exp(2*x))-1)*exp(2)), x)

$\label{eq9}\frac{{{e}^{2}}\ {{e}^{x + 2}}}{{{{x}^{2}}\ {{e}^{2}}\ {\log \left({2 \ x}\right)}}+{{e}^{x + 2}}}$

(9)

Type: Union(Expression(Integer),...)

I see that Axiom failed to recognize that e^4 - (e^2)^2 is zero. AFAICS similar problem appears in original example: when I trace rischNormalize I see that both e^2 and e^4 appear in "normalized" function. But this may cause trouble later, since Axiom assumes that different kernels are algebraically independent. I do not know if rischNormalize is supposed to simplify constants (maybe this shoud be done by a separtate pass) but failing to recognize "transcendental" zero may give wrong results and otherwise Risch algorithm may declare an integrable function as non integrable.

fixed in FriCAS revision 475 --kratt6, Fri, 10 Apr 2009 03:20:47 -0700 reply

Status: open => fixed somewhere

... --test1, Thu, 08 Feb 2018 16:23:52 +0000 reply

Status: fixed somewhere => closed

Subject: Be Bold !!

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