\[ \int x^3 \cosh ^{-1}(a x)^4 \, dx \]
Optimal antiderivative \[ \frac {45 x^{2}}{128 a^{2}}+\frac {3 x^{4}}{128}-\frac {45 \mathrm {arccosh}\! \left (a x \right )^{2}}{128 a^{4}}+\frac {9 x^{2} \mathrm {arccosh}\! \left (a x \right )^{2}}{16 a^{2}}+\frac {3 x^{4} \mathrm {arccosh}\! \left (a x \right )^{2}}{16}-\frac {3 \mathrm {arccosh}\! \left (a x \right )^{4}}{32 a^{4}}+\frac {x^{4} \mathrm {arccosh}\! \left (a x \right )^{4}}{4}-\frac {45 x \,\mathrm {arccosh}\! \left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}}{64 a^{3}}-\frac {3 x^{3} \mathrm {arccosh}\! \left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}}{32 a}-\frac {3 x \mathrm {arccosh}\! \left (a x \right )^{3} \sqrt {a x -1}\, \sqrt {a x +1}}{8 a^{3}}-\frac {x^{3} \mathrm {arccosh}\! \left (a x \right )^{3} \sqrt {a x -1}\, \sqrt {a x +1}}{4 a} \]
command
int(x^3*arccosh(a*x)^4,x)
Maple 2022.1 output
\[\int x^{3} \mathrm {arccosh}\left (a x \right )^{4}\, dx\]
Maple 2021.1 output
\[ \frac {\frac {a^{4} x^{4} \mathrm {arccosh}\left (a x \right )^{4}}{4}-\frac {\mathrm {arccosh}\left (a x \right )^{3} \sqrt {a x -1}\, \sqrt {a x +1}\, a^{3} x^{3}}{4}-\frac {3 \mathrm {arccosh}\left (a x \right )^{3} \sqrt {a x -1}\, \sqrt {a x +1}\, a x}{8}-\frac {3 \mathrm {arccosh}\left (a x \right )^{4}}{32}+\frac {3 a^{4} x^{4} \mathrm {arccosh}\left (a x \right )^{2}}{16}-\frac {3 \,\mathrm {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}\, a^{3} x^{3}}{32}-\frac {45 \,\mathrm {arccosh}\left (a x \right ) a x \sqrt {a x -1}\, \sqrt {a x +1}}{64}-\frac {45 \mathrm {arccosh}\left (a x \right )^{2}}{128}+\frac {3 x^{4} a^{4}}{128}+\frac {45 a^{2} x^{2}}{128}+\frac {9 a^{2} x^{2} \mathrm {arccosh}\left (a x \right )^{2}}{16}}{a^{4}} \]