 Symbolic Integration - Maple Programming Help

 Symbolic Integration

Indefinite integration

The capabilities of finding indefinite integrals in Maple have been improved. The following integrals could not be computed in previous versions of Maple. In particular, this applies to many integrands involving inverse hyperbolic functions, such as the following:

 ${-}\frac{{1}}{{2}}{}{{\mathrm{arcsinh}}{}\left({x}\right)}^{{2}}{+}{\mathrm{arcsinh}}{}\left({x}\right){}{\mathrm{ln}}{}\left({1}{-}{x}{-}\sqrt{{{x}}^{{2}}{+}{1}}\right){+}{\mathrm{polylog}}{}\left({2}{,}{x}{+}\sqrt{{{x}}^{{2}}{+}{1}}\right){+}{\mathrm{arcsinh}}{}\left({x}\right){}{\mathrm{ln}}{}\left({1}{+}{x}{+}\sqrt{{{x}}^{{2}}{+}{1}}\right){+}{\mathrm{polylog}}{}\left({2}{,}{-}{x}{-}\sqrt{{{x}}^{{2}}{+}{1}}\right)$ (1.1)

$\frac{ⅆ}{ⅆx}$

 ${-}\frac{{\mathrm{arcsinh}}{}\left({x}\right)}{\sqrt{{{x}}^{{2}}{+}{1}}}{+}\frac{{\mathrm{ln}}{}\left({1}{-}{x}{-}\sqrt{{{x}}^{{2}}{+}{1}}\right)}{\sqrt{{{x}}^{{2}}{+}{1}}}{+}\frac{{\mathrm{arcsinh}}{}\left({x}\right){}\left({-}{1}{-}\frac{{x}}{\sqrt{{{x}}^{{2}}{+}{1}}}\right)}{{1}{-}{x}{-}\sqrt{{{x}}^{{2}}{+}{1}}}{-}\frac{\left({1}{+}\frac{{x}}{\sqrt{{{x}}^{{2}}{+}{1}}}\right){}{\mathrm{ln}}{}\left({1}{-}{x}{-}\sqrt{{{x}}^{{2}}{+}{1}}\right)}{{x}{+}\sqrt{{{x}}^{{2}}{+}{1}}}{+}\frac{{\mathrm{ln}}{}\left({1}{+}{x}{+}\sqrt{{{x}}^{{2}}{+}{1}}\right)}{\sqrt{{{x}}^{{2}}{+}{1}}}{+}\frac{{\mathrm{arcsinh}}{}\left({x}\right){}\left({1}{+}\frac{{x}}{\sqrt{{{x}}^{{2}}{+}{1}}}\right)}{{1}{+}{x}{+}\sqrt{{{x}}^{{2}}{+}{1}}}{-}\frac{\left({-}{1}{-}\frac{{x}}{\sqrt{{{x}}^{{2}}{+}{1}}}\right){}{\mathrm{ln}}{}\left({1}{+}{x}{+}\sqrt{{{x}}^{{2}}{+}{1}}\right)}{{-}{x}{-}\sqrt{{{x}}^{{2}}{+}{1}}}$ (1.2)

$\mathrm{radnormal}\left(\right)$

 $\frac{{\mathrm{arcsinh}}{}\left({x}\right)}{{x}}$ (1.3)

$\int \mathrm{arccoth}{\left(x\right)}^{3}ⅆx$

 ${{\mathrm{arccoth}}{}\left({x}\right)}^{{3}}{}\left({x}{-}{1}\right){+}{2}{}{{\mathrm{arccoth}}{}\left({x}\right)}^{{3}}{-}{3}{}{{\mathrm{arccoth}}{}\left({x}\right)}^{{2}}{}{\mathrm{ln}}{}\left({1}{-}\frac{{1}}{\sqrt{\frac{{x}{-}{1}}{{x}{+}{1}}}}\right){-}{6}{}{\mathrm{arccoth}}{}\left({x}\right){}{\mathrm{polylog}}{}\left({2}{,}\frac{{1}}{\sqrt{\frac{{x}{-}{1}}{{x}{+}{1}}}}\right){+}{6}{}{\mathrm{polylog}}{}\left({3}{,}\frac{{1}}{\sqrt{\frac{{x}{-}{1}}{{x}{+}{1}}}}\right){-}{3}{}{{\mathrm{arccoth}}{}\left({x}\right)}^{{2}}{}{\mathrm{ln}}{}\left({1}{+}\frac{{1}}{\sqrt{\frac{{x}{-}{1}}{{x}{+}{1}}}}\right){-}{6}{}{\mathrm{arccoth}}{}\left({x}\right){}{\mathrm{polylog}}{}\left({2}{,}{-}\frac{{1}}{\sqrt{\frac{{x}{-}{1}}{{x}{+}{1}}}}\right){+}{6}{}{\mathrm{polylog}}{}\left({3}{,}{-}\frac{{1}}{\sqrt{\frac{{x}{-}{1}}{{x}{+}{1}}}}\right)$ (1.4)

$\mathrm{radnormal}\left(\frac{ⅆ}{ⅆx}\right)$

 ${{\mathrm{arccoth}}{}\left({x}\right)}^{{3}}$ (1.5)

 ${\mathrm{ln}}{}\left({x}\right){}{{\mathrm{arctanh}}{}\left({\mathrm{tanh}}{}\left({b}{}{x}{+}{a}\right)\right)}^{{2}}{+}{{b}}^{{2}}{}{{x}}^{{2}}{}{\mathrm{ln}}{}\left({x}\right){-}\frac{{3}}{{2}}{}{{b}}^{{2}}{}{{x}}^{{2}}{-}{2}{}{b}{}{\mathrm{ln}}{}\left({x}\right){}{\mathrm{arctanh}}{}\left({\mathrm{tanh}}{}\left({b}{}{x}{+}{a}\right)\right){}{x}{+}{2}{}{b}{}{\mathrm{arctanh}}{}\left({\mathrm{tanh}}{}\left({b}{}{x}{+}{a}\right)\right){}{x}$ (1.6)

$\frac{ⅆ}{ⅆx}$

 $\frac{{{\mathrm{arctanh}}{}\left({\mathrm{tanh}}{}\left({b}{}{x}{+}{a}\right)\right)}^{{2}}}{{x}}$ (1.7)

Some other types of integrands are covered by the improvements as well.

$\int \frac{{\mathrm{ln}\left({\left(x+a\right)}^{n}\right)}^{2}}{{\left(x+b\right)}^{2}}ⅆx$

 ${-}\frac{{{\mathrm{ln}}{}\left({\left({x}{+}{a}\right)}^{{n}}\right)}^{{2}}}{{x}{+}{b}}{+}\frac{{2}{}{n}{}{\mathrm{ln}}{}\left({\left({x}{+}{a}\right)}^{{n}}\right){}{\mathrm{ln}}{}\left({x}{+}{b}\right)}{{a}{-}{b}}{-}\frac{{2}{}{n}{}{\mathrm{ln}}{}\left({\left({x}{+}{a}\right)}^{{n}}\right){}{\mathrm{ln}}{}\left({x}{+}{a}\right)}{{a}{-}{b}}{-}\frac{{2}{}{{n}}^{{2}}{}{\mathrm{ln}}{}\left({x}{+}{b}\right){}{\mathrm{ln}}{}\left(\frac{{x}{+}{a}}{{a}{-}{b}}\right)}{{a}{-}{b}}{-}\frac{{2}{}{{n}}^{{2}}{}{\mathrm{dilog}}{}\left(\frac{{x}{+}{a}}{{a}{-}{b}}\right)}{{a}{-}{b}}{+}\frac{{{n}}^{{2}}{}{{\mathrm{ln}}{}\left({x}{+}{a}\right)}^{{2}}}{{a}{-}{b}}$ (1.8)

$\mathrm{normal}\left(\frac{ⅆ}{ⅆx}\right)$

 $\frac{{{\mathrm{ln}}{}\left({\left({x}{+}{a}\right)}^{{n}}\right)}^{{2}}}{{\left({x}{+}{b}\right)}^{{2}}}$ (1.9)



More compact results

Some integrals that used to be expressed in terms of lengthy csgn expressions are now are given in more compact form.

 $\frac{{1}}{{2}}{}{{x}}^{{2}}{}{{\mathrm{arctan}}{}\left({\mathrm{tan}}{}\left({x}\right)\right)}^{{3}}{-}\frac{{1}}{{2}}{}{{x}}^{{3}}{}{{\mathrm{arctan}}{}\left({\mathrm{tan}}{}\left({x}\right)\right)}^{{2}}{+}\frac{{1}}{{4}}{}{{x}}^{{4}}{}{\mathrm{arctan}}{}\left({\mathrm{tan}}{}\left({x}\right)\right){-}\frac{{1}}{{20}}{}{{x}}^{{5}}$ (1.1.1)

$\frac{ⅆ}{ⅆx}$

 ${x}{}{{\mathrm{arctan}}{}\left({\mathrm{tan}}{}\left({x}\right)\right)}^{{3}}$ (1.1.2)



Definite integration

Definite integrals can now also be computed for some non-smooth integrands, for which previous versions of Maple could only compute an indefinite integral.

${\int }_{-\mathrm{∞}}^{\mathrm{∞}}\frac{{\left|x-1\right|}^{1/3}}{{x}^{2}+1}ⅆx$

 ${-}{3}{}\left({\sum }_{{\mathrm{_R}}{=}{\mathrm{RootOf}}{}\left({23328}{}{{\mathrm{_Z}}}^{{6}}{+}{216}{}{{\mathrm{_Z}}}^{{3}}{+}{1}\right)}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({\mathrm{_R}}{}{\mathrm{ln}}{}\left({6}\right){+}{\mathrm{_R}}{}{\mathrm{ln}}{}\left({-}{216}{}{{\mathrm{_R}}}^{{4}}{-}{\mathrm{_R}}\right)\right)\right){-}{3}{}\left({\sum }_{{\mathrm{_R}}{=}{\mathrm{RootOf}}{}\left({23328}{}{{\mathrm{_Z}}}^{{6}}{+}{216}{}{{\mathrm{_Z}}}^{{3}}{+}{1}\right)}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({\mathrm{_R}}{}{\mathrm{ln}}{}\left({6}\right){+}{\mathrm{_R}}{}{\mathrm{ln}}{}\left({216}{}{{\mathrm{_R}}}^{{4}}{+}{\mathrm{_R}}\right)\right)\right)$ (2.1)

$\mathrm{radnormal}\left(\mathrm{evalc}\left(\mathrm{allvalues}\left(\right)\right)\right)$

 $\frac{{1}}{{6}}{}{{2}}^{{2}{/}{3}}{}{\mathrm{π}}{}\left({3}{+}\sqrt{{3}}\right)$ (2.2)