\[ \int x^2 \cosh ^{-1}(a x)^3 \, dx \]
Optimal antiderivative \[ \frac {4 x \,\mathrm {arccosh}\! \left (a x \right )}{3 a^{2}}+\frac {2 x^{3} \mathrm {arccosh}\! \left (a x \right )}{9}+\frac {x^{3} \mathrm {arccosh}\! \left (a x \right )^{3}}{3}-\frac {40 \sqrt {a x -1}\, \sqrt {a x +1}}{27 a^{3}}-\frac {2 x^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{27 a}-\frac {2 \mathrm {arccosh}\! \left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{3 a^{3}}-\frac {x^{2} \mathrm {arccosh}\! \left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{3 a} \]
command
int(x^2*arccosh(a*x)^3,x)
Maple 2022.1 output
\[\int x^{2} \mathrm {arccosh}\left (a x \right )^{3}\, dx\]
Maple 2021.1 output
\[ \frac {\frac {a^{3} x^{3} \mathrm {arccosh}\left (a x \right )^{3}}{3}-\frac {2 \mathrm {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{3}-\frac {\mathrm {arccosh}\left (a x \right )^{2} a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{3}+\frac {4 a x \,\mathrm {arccosh}\left (a x \right )}{3}-\frac {40 \sqrt {a x -1}\, \sqrt {a x +1}}{27}+\frac {2 a^{3} x^{3} \mathrm {arccosh}\left (a x \right )}{9}-\frac {2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{27}}{a^{3}} \]