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ElementaryIntegrationExample ←	↑	→ FriCASTimofeev1

FriCASSpecialIntegration

Edit detail for FriCASSpecialIntegration revision 3 of 6

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Editor: test1 Time: 2017/12/27 16:44:21 GMT+0
Note:

added:
Since FriCAS uses algorithmic approach some integrals can be done easily without any extra special
support.  For example:
\begin{axiom}
)set output tex off
)set output algebra on
integrate(-erf(((2*m - k^2) - 2*log(c + b) + 2*log(a))/(2*sqrt(2)*k))/2 - 1/2, c)
)set output tex on
)set output algebra off
\end{axiom}
is done combining general support for Liouvillian integrands with procedure for handling erf.
In Rubi this example required adding a new special rule.

FriCAS? can now handle large class of integrals expressible in terms of exponential integral, error functions, incomplete Gamma function with rational first argument, 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

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

$\label{eq12}-{{{\erf \left({{x + 1}\over x}\right)}\ {\sqrt{\pi}}}\over 2}$

(12)

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

fricas

integrate(x^3*exp(-x^3), x)

$\label{eq13}{-{\Gamma \left({{1 \over 3}, \:{{x}^{3}}}\right)}-{3 \ x \ {{e}^{-{{x}^{3}}}}}}\over 9$

(13)

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

fricas

integrate(x^2*exp(-(x+1)^3), x)

$\label{eq14}{\left( \begin{array}{@{}l} \displaystyle {2 \ {\Gamma \left({{2 \over 3}, \:{{{x}^{3}}+{3 \ {{x}^{2}}}+{3 \ x}+ 1}}\right)}}- \ \ \displaystyle {\Gamma \left({{1 \over 3}, \:{{{x}^{3}}+{3 \ {{x}^{2}}}+{3 \ x}+ 1}}\right)}- \ \ \displaystyle {{e}^{-{{x}^{3}}-{3 \ {{x}^{2}}}-{3 \ x}- 1}}$

(14)

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

FriCAS? can introduce new algebraic constants when needed:

fricas

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

$\label{eq15}{-{li \left({x \ {{e}^{\sqrt{3}}}}\right)}+{{{{e}^{\sqrt{3}}}^{2}}\ {li \left({x \over{{e}^{\sqrt{3}}}}\right)}}}\over{2 \ {\sqrt{3}}\ {{e}^{\sqrt{3}}}}$

(15)

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

fricas

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

$\label{eq16}{{{Ei \left({-{\sqrt{5}}+ x}\right)}\ {{{e}^{\sqrt{5}}}^{2}}}-{Ei \left({{\sqrt{5}}+ x}\right)}}\over{2 \ {\sqrt{5}}\ {{e}^{\sqrt{5}}}}$

(16)

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

The method is robust, FriCAS? can handle both

fricas

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

$\label{eq17}li \left({x \ {{e}^{x}}}\right)$

(17)

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

fricas

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

$\label{eq18}li \left({x \ {{e}^{x}}}\right)$

(18)

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)

$\label{eq19}\begin{array}{@{}l} \displaystyle {{\left(-{2 \ {\log \left({x}\right)}}- x \right)}\ {\log \left({{{3 \ {{e}^{{2 \ {\log \left({x}\right)}}+ x}}}+{\sqrt{3}}}\over{\sqrt{3}}}\right)}}+ \ \ \displaystyle {{\left(-{2 \ {\log \left({x}\right)}}- x \right)}\ {\log \left({{-{3 \ {{e}^{{2 \ {\log \left({x}\right)}}+ x}}}+{\sqrt{3}}}\over{\sqrt{3}}}\right)}}- \ \ \displaystyle {dilog \left({{{3 \ {{e}^{{2 \ {\log \left({x}\right)}}+ x}}}+{\sqrt{3}}}\over{\sqrt{3}}}\right)}- \ \ \displaystyle {dilog \left({{-{3 \ {{e}^{{2 \ {\log \left({x}\right)}}+ x}}}+{\sqrt{3}}}\over{\sqrt{3}}}\right)}+{4 \ {{\log \left({x}\right)}^{2}}}+{4 \ x \ {\log \left({x}\right)}}+ \ \ \displaystyle {{x}^{2}}$

(19)

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)

$\label{eq20}\begin{array}{@{}l} \displaystyle {{\left(-{2 \ {\log \left({x}\right)}}- x \right)}\ {\log \left({{{3 \ {{e}^{{2 \ {\log \left({x}\right)}}+ x}}}+{\sqrt{3}}}\over{\sqrt{3}}}\right)}}+ \ \ \displaystyle {{\left(-{2 \ {\log \left({x}\right)}}- x \right)}\ {\log \left({{-{3 \ {{e}^{{2 \ {\log \left({x}\right)}}+ x}}}+{\sqrt{3}}}\over{\sqrt{3}}}\right)}}- \ \ \displaystyle {dilog \left({{{3 \ {{e}^{{2 \ {\log \left({x}\right)}}+ x}}}+{\sqrt{3}}}\over{\sqrt{3}}}\right)}- \ \ \displaystyle {dilog \left({{-{3 \ {{e}^{{2 \ {\log \left({x}\right)}}+ x}}}+{\sqrt{3}}}\over{\sqrt{3}}}\right)}+{4 \ {{\log \left({x}\right)}^{2}}}+{4 \ x \ {\log \left({x}\right)}}+ \ \ \displaystyle {{x}^{2}}$

(20)

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)

$\label{eq21}{{\left({{x}^{2}}+ 2 \right)}\ {{e}^{x \over{{{x}^{2}}+ 2}}}}+{Ei \left({x \over{{{x}^{2}}+ 2}}\right)}$

(21)

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

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

Since FriCAS? uses algorithmic approach some integrals can be done easily without any extra special support. For example:

fricas

)set output tex off

fricas

)set output algebra on

integrate(-erf(((2*m - k^2) - 2*log(c + b) + 2*log(a))/(2*sqrt(2)*k))/2 - 1/2, c)

   (22)
                                                   +---+
                                                2  | 1
                 (2log(c + b) - 2log(a) - 2m - k ) |---
                                                   |  2
          +-+                                     \|2k     log(a) + m
       - \|2 erf(---------------------------------------)%e
                                    2
     + 
                                                  +---+
                                               2  | 1
                (2log(c + b) - 2log(a) - 2m + k ) |---
                                                  |  2
         +-+                                     \|2k
       b\|2 erf(---------------------------------------)
                                   2
     + 
            +---+                                  2         +---+
            | 1      2log(c + b) - 2log(a) - 2m + k          | 1
       2c k |--- erf(-------------------------------) - 2c k |---
            |  2                     +-+                     |  2
           \|2k                   2k\|2                     \|2k
  /
        +---+
        | 1
     4k |---
        |  2
       \|2k

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

fricas

)set output tex on

fricas

)set output algebra off

is done combining general support for Liouvillian integrands with procedure for handling erf. In Rubi this example required adding a new special rule.

FriCAS? can also handle some integrals involving special functions of algebraic arguments:

fricas

integrate(((26*x+23)*x^(1/2)+4*x^2+50*x-6)*exp(2*x^(1/2)+x)/((16*x^2+36*x)*x^(1/2)+(2*x^3+42*x^2)), x)

$\label{eq22}{{{e}^{{2 \ {\sqrt{x}}}+ x}}+{{\left({3 \ {\sqrt{x}}}+ x \right)}\ {Ei \left({{2 \ {\sqrt{x}}}+ x}\right)}}}\over{{3 \ {\sqrt{x}}}+ x}$

(22)

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