\[ \int x^4 \cosh ^{-1}(a x)^3 \, dx \]
Optimal antiderivative \[ \frac {16 x \,\mathrm {arccosh}\! \left (a x \right )}{25 a^{4}}+\frac {8 x^{3} \mathrm {arccosh}\! \left (a x \right )}{75 a^{2}}+\frac {6 x^{5} \mathrm {arccosh}\! \left (a x \right )}{125}+\frac {x^{5} \mathrm {arccosh}\! \left (a x \right )^{3}}{5}-\frac {4144 \sqrt {a x -1}\, \sqrt {a x +1}}{5625 a^{5}}-\frac {272 x^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{5625 a^{3}}-\frac {6 x^{4} \sqrt {a x -1}\, \sqrt {a x +1}}{625 a}-\frac {8 \mathrm {arccosh}\! \left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{25 a^{5}}-\frac {4 x^{2} \mathrm {arccosh}\! \left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{25 a^{3}}-\frac {3 x^{4} \mathrm {arccosh}\! \left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{25 a} \]
command
int(x^4*arccosh(a*x)^3,x)
Maple 2022.1 output
\[\int x^{4} \mathrm {arccosh}\left (a x \right )^{3}\, dx\]
Maple 2021.1 output
\[ \frac {\frac {a^{5} x^{5} \mathrm {arccosh}\left (a x \right )^{3}}{5}-\frac {8 \mathrm {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{25}-\frac {3 \mathrm {arccosh}\left (a x \right )^{2} a^{4} x^{4} \sqrt {a x -1}\, \sqrt {a x +1}}{25}-\frac {4 \mathrm {arccosh}\left (a x \right )^{2} a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{25}+\frac {16 a x \,\mathrm {arccosh}\left (a x \right )}{25}-\frac {4144 \sqrt {a x -1}\, \sqrt {a x +1}}{5625}+\frac {6 a^{5} x^{5} \mathrm {arccosh}\left (a x \right )}{125}-\frac {6 a^{4} x^{4} \sqrt {a x -1}\, \sqrt {a x +1}}{625}-\frac {272 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{5625}+\frac {8 a^{3} x^{3} \mathrm {arccosh}\left (a x \right )}{75}}{a^{5}} \]