\[ \int x^3 \cosh ^{-1}(a x)^n \, dx \]
Optimal antiderivative \[ \frac {\mathrm {arccosh}\left (a x \right )^{n} \Gamma \left (1+n , -4 \,\mathrm {arccosh}\left (a x \right )\right ) 2^{-2 n} \left (-\mathrm {arccosh}\left (a x \right )\right )^{-n}}{64 a^{4}}+\frac {2^{-4-n} \mathrm {arccosh}\left (a x \right )^{n} \Gamma \left (1+n , -2 \,\mathrm {arccosh}\left (a x \right )\right ) \left (-\mathrm {arccosh}\left (a x \right )\right )^{-n}}{a^{4}}+\frac {2^{-4-n} \Gamma \left (1+n , 2 \,\mathrm {arccosh}\left (a x \right )\right )}{a^{4}}+\frac {\Gamma \left (1+n , 4 \,\mathrm {arccosh}\left (a x \right )\right ) 2^{-2 n}}{64 a^{4}} \]
command
int(x^3*arccosh(a*x)^n,x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| default | \(\frac {\mathrm {arccosh}\left (a x \right )^{n +2} \hypergeom \left (\left [1+\frac {n}{2}\right ], \left [\frac {3}{2}, 2+\frac {n}{2}\right ], \mathrm {arccosh}\left (a x \right )^{2}\right )}{2 a^{4} \left (n +2\right )}+\frac {\mathrm {arccosh}\left (a x \right )^{n +2} \hypergeom \left (\left [1+\frac {n}{2}\right ], \left [\frac {3}{2}, 2+\frac {n}{2}\right ], 4 \mathrm {arccosh}\left (a x \right )^{2}\right )}{2 a^{4} \left (n +2\right )}\) | \(80\) |
Maple 2021.1 output
\[ \int x^{3} \mathrm {arccosh}\left (a x \right )^{n}\, dx \]________________________________________________________________________________________