\[ \int x^3 \cosh ^{-1}(a x)^3 \, dx \]
Optimal antiderivative \[ -\frac {45 \,\mathrm {arccosh}\! \left (a x \right )}{256 a^{4}}+\frac {9 x^{2} \mathrm {arccosh}\! \left (a x \right )}{32 a^{2}}+\frac {3 x^{4} \mathrm {arccosh}\! \left (a x \right )}{32}-\frac {3 \mathrm {arccosh}\! \left (a x \right )^{3}}{32 a^{4}}+\frac {x^{4} \mathrm {arccosh}\! \left (a x \right )^{3}}{4}-\frac {45 x \sqrt {a x -1}\, \sqrt {a x +1}}{256 a^{3}}-\frac {3 x^{3} \sqrt {a x -1}\, \sqrt {a x +1}}{128 a}-\frac {9 x \mathrm {arccosh}\! \left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{32 a^{3}}-\frac {3 x^{3} \mathrm {arccosh}\! \left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{16 a} \]
command
int(x^3*arccosh(a*x)^3,x)
Maple 2022.1 output
\[\int x^{3} \mathrm {arccosh}\left (a x \right )^{3}\, dx\]
Maple 2021.1 output
\[ \frac {\frac {a^{4} x^{4} \mathrm {arccosh}\left (a x \right )^{3}}{4}-\frac {3 \mathrm {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}\, a^{3} x^{3}}{16}-\frac {9 \mathrm {arccosh}\left (a x \right )^{2} a x \sqrt {a x -1}\, \sqrt {a x +1}}{32}-\frac {3 \mathrm {arccosh}\left (a x \right )^{3}}{32}+\frac {3 a^{4} x^{4} \mathrm {arccosh}\left (a x \right )}{32}-\frac {3 a^{3} x^{3} \sqrt {a x -1}\, \sqrt {a x +1}}{128}-\frac {45 \sqrt {a x +1}\, \sqrt {a x -1}\, a x}{256}-\frac {45 \,\mathrm {arccosh}\left (a x \right )}{256}+\frac {9 a^{2} x^{2} \mathrm {arccosh}\left (a x \right )}{32}}{a^{4}} \]