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	FriCASIntegration
ElementaryIntegrationExample ←	↑	→ FriCASTimofeev1

FriCASSpecialIntegration

Edit detail for FriCASSpecialIntegration revision 1 of 6

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Editor: test1 Time: 2014/04/04 15:57:02 GMT+0
Note:

changed:
-
FriCAS can now handle large class of integrals expressible in terms of exponential
integral, logarithmic integral and polylogarithms.  Like
\begin{axiom}
integrate(1/log(x), x)
integrate(1/(log(x) + 1), x)
integrate(1/(log(x)^2-1), x)
integrate(exp(x + a)/x, x)
integrate(exp(x + a)/x^2, x)
integrate(exp(x)/(x^2 - 1), x)
integrate(x/(exp(x) - 1), x)
integrate(x^3/(exp(x) - 1), x)
integrate(2*x*exp(x)/(exp(x)^2 - 1), x)
integrate(x/sinh(x), x)
integrate(log(sinh(x)), x)
\end{axiom}

FriCAS can introduce new algebraic constants when needed:
\begin{axiom}
integrate(1/(log(x)^2-3), x)
integrate(exp(x)/(x^2 - 5), x)
\end{axiom}

The method is robust, FriCAS can handle both
\begin{axiom}
integrate(((x+1)*exp(x))/log(x*exp(x)), x)
integrate(((x+1)*exp(x))/(x + log(x)), x)
\end{axiom}
while Mathematca 8 can handle the first form, but not the second one
(Maple 15 and Maxima 5.30.0 can not handle any).

Similarly FriCAS has no troubles with
\begin{axiom}
integrate(((-4*x-8)*log(x)+(-2*x^2-4*x))/(3*x*exp(2*log(x)+x)^2-x), x)
integrate(((-4*x-8)*log(x)+(-2*x^2-4*x))/(3*x^3*exp(log(x)+x)^2-x), x)
integrate(((2*x^4-x^3+3*x^2+2*x+2)*exp(x/(x^2+2)))/(x^3+2*x), x)
\end{axiom}
none of Mathematca 8, Maple 15 and Maxima 5.30.0 can handle them.

FriCAS? can now handle large class of integrals expressible in terms of exponential integral, logarithmic integral and polylogarithms. Like

fricas

integrate(1/log(x), x)

$\label{eq1}li \left({x}\right)$

(1)

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

fricas

integrate(1/(log(x) + 1), x)

$\label{eq2}{li \left({x \ e}\right)}\over e$

(2)

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

fricas

integrate(1/(log(x)^2-1), x)

$\label{eq3}{-{li \left({x \ e}\right)}+{{{e}^{2}}\ {li \left({x \over e}\right)}}}\over{2 \ e}$

(3)

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

fricas

integrate(exp(x + a)/x, x)

$\label{eq4}{Ei \left({x}\right)}\ {{e}^{a}}$

(4)

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

fricas

integrate(exp(x + a)/x^2, x)

$\label{eq5}{-{{e}^{x + a}}+{x \ {Ei \left({x}\right)}\ {{e}^{a}}}}\over x$

(5)

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

fricas

integrate(exp(x)/(x^2 - 1), x)

$\label{eq6}{-{Ei \left({x + 1}\right)}+{{{e}^{2}}\ {Ei \left({x - 1}\right)}}}\over{2 \ e}$

(6)

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

fricas

integrate(x/(exp(x) - 1), x)

$\label{eq7}{{2 \ x \ {\log \left({-{{e}^{x}}+ 1}\right)}}+{2 \ {dilog \left({-{{e}^{x}}+ 1}\right)}}-{{x}^{2}}}\over 2$

(7)

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

fricas

integrate(x^3/(exp(x) - 1), x)

$\label{eq8}{\left( \begin{array}{@{}l} \displaystyle {{24}\ {polylog \left({4, \:{{e}^{x}}}\right)}}-{{24}\ x \ {polylog \left({3, \:{{e}^{x}}}\right)}}+ \ \ \displaystyle {4 \ {{x}^{3}}\ {\log \left({-{{e}^{x}}+ 1}\right)}}+{{12}\ {{x}^{2}}\ {dilog \left({-{{e}^{x}}+ 1}\right)}}-{{x}^{4}}$

(8)

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

fricas

integrate(2*x*exp(x)/(exp(x)^2 - 1), x)

$\label{eq9}\begin{array}{@{}l} \displaystyle -{x \ {\log \left({{{e}^{x}}+ 1}\right)}}+{x \ {\log \left({-{{e}^{x}}+ 1}\right)}}-{dilog \left({{{e}^{x}}+ 1}\right)}+ \ \ \displaystyle {dilog \left({-{{e}^{x}}+ 1}\right)}$

(9)

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

fricas

integrate(x/sinh(x), x)

$\label{eq10}\begin{array}{@{}l} \displaystyle -{x \ {\log \left({{\sinh \left({x}\right)}+{\cosh \left({x}\right)}+ 1}\right)}}+ \ \ \displaystyle {x \ {\log \left({-{\sinh \left({x}\right)}-{\cosh \left({x}\right)}+ 1}\right)}}- \ \ \displaystyle {dilog \left({{\sinh \left({x}\right)}+{\cosh \left({x}\right)}+ 1}\right)}+ \ \ \displaystyle {dilog \left({-{\sinh \left({x}\right)}-{\cosh \left({x}\right)}+ 1}\right)}$

(10)

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

fricas

integrate(log(sinh(x)), x)

$\label{eq11}{\left( \begin{array}{@{}l} \displaystyle -{2 \ x \ {\log \left({{\sinh \left({x}\right)}+{\cosh \left({x}\right)}+ 1}\right)}}+{2 \ x \ {\log \left({\sinh \left({x}\right)}\right)}}- \ \ \displaystyle {2 \ x \ {\log \left({-{\sinh \left({x}\right)}-{\cosh \left({x}\right)}+ 1}\right)}}- \ \ \displaystyle {2 \ {dilog \left({{\sinh \left({x}\right)}+{\cosh \left({x}\right)}+ 1}\right)}}- \ \ \displaystyle {2 \ {dilog \left({-{\sinh \left({x}\right)}-{\cosh \left({x}\right)}+ 1}\right)}}+{{x}^{2}}$

(11)

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

FriCAS? can introduce new algebraic constants when needed:

fricas

integrate(1/(log(x)^2-3), x)

$\label{eq12}{-{li \left({x \ {{e}^{\sqrt{3}}}}\right)}+{{{{e}^{\sqrt{3}}}^{2}}\ {li \left({x \over{{e}^{\sqrt{3}}}}\right)}}}\over{2 \ {\sqrt{3}}\ {{e}^{\sqrt{3}}}}$

(12)

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

fricas

integrate(exp(x)/(x^2 - 5), x)

$\label{eq13}{{{Ei \left({-{\sqrt{5}}+ x}\right)}\ {{{e}^{\sqrt{5}}}^{2}}}-{Ei \left({{\sqrt{5}}+ x}\right)}}\over{2 \ {\sqrt{5}}\ {{e}^{\sqrt{5}}}}$

(13)

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

The method is robust, FriCAS? can handle both

fricas

integrate(((x+1)*exp(x))/log(x*exp(x)), x)

$\label{eq14}li \left({x \ {{e}^{x}}}\right)$

(14)

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

fricas

integrate(((x+1)*exp(x))/(x + log(x)), x)

$\label{eq15}li \left({x \ {{e}^{x}}}\right)$

(15)

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

while Mathematca 8 can handle the first form, but not the second one (Maple 15 and Maxima 5.30.0 can not handle any).

Similarly FriCAS? has no troubles with

fricas

integrate(((-4*x-8)*log(x)+(-2*x^2-4*x))/(3*x*exp(2*log(x)+x)^2-x), x)

$\label{eq16}\begin{array}{@{}l} \displaystyle {{\left(-{2 \ {\log \left({x}\right)}}- x \right)}\ {\log \left({{{3 \ {{e}^{{2 \ {\log \left({x}\right)}}+ x}}}+{\sqrt{3}}}\over{\sqrt{3}}}\right)}}+ \ \ \displaystyle {{\left(-{2 \ {\log \left({x}\right)}}- x \right)}\ {\log \left({{-{3 \ {{e}^{{2 \ {\log \left({x}\right)}}+ x}}}+{\sqrt{3}}}\over{\sqrt{3}}}\right)}}- \ \ \displaystyle {dilog \left({{{3 \ {{e}^{{2 \ {\log \left({x}\right)}}+ x}}}+{\sqrt{3}}}\over{\sqrt{3}}}\right)}- \ \ \displaystyle {dilog \left({{-{3 \ {{e}^{{2 \ {\log \left({x}\right)}}+ x}}}+{\sqrt{3}}}\over{\sqrt{3}}}\right)}+{4 \ {{\log \left({x}\right)}^{2}}}+{4 \ x \ {\log \left({x}\right)}}+ \ \ \displaystyle {{x}^{2}}$

(16)

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

fricas

integrate(((-4*x-8)*log(x)+(-2*x^2-4*x))/(3*x^3*exp(log(x)+x)^2-x), x)

$\label{eq17}\begin{array}{@{}l} \displaystyle {{\left(-{2 \ {\log \left({x}\right)}}- x \right)}\ {\log \left({{{3 \ {{e}^{{2 \ {\log \left({x}\right)}}+ x}}}+{\sqrt{3}}}\over{\sqrt{3}}}\right)}}+ \ \ \displaystyle {{\left(-{2 \ {\log \left({x}\right)}}- x \right)}\ {\log \left({{-{3 \ {{e}^{{2 \ {\log \left({x}\right)}}+ x}}}+{\sqrt{3}}}\over{\sqrt{3}}}\right)}}- \ \ \displaystyle {dilog \left({{{3 \ {{e}^{{2 \ {\log \left({x}\right)}}+ x}}}+{\sqrt{3}}}\over{\sqrt{3}}}\right)}- \ \ \displaystyle {dilog \left({{-{3 \ {{e}^{{2 \ {\log \left({x}\right)}}+ x}}}+{\sqrt{3}}}\over{\sqrt{3}}}\right)}+{4 \ {{\log \left({x}\right)}^{2}}}+{4 \ x \ {\log \left({x}\right)}}+ \ \ \displaystyle {{x}^{2}}$

(17)

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

fricas

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

$\label{eq18}{{\left({{x}^{2}}+ 2 \right)}\ {{e}^{x \over{{{x}^{2}}+ 2}}}}+{Ei \left({x \over{{{x}^{2}}+ 2}}\right)}$

(18)

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

none of Mathematca 8, Maple 15 and Maxima 5.30.0 can handle them.