\[ \int x^3 \cosh ^{-1}(a x)^{3/2} \, dx \]
Optimal antiderivative \[ -\frac {3 \mathrm {arccosh}\left (a x \right )^{\frac {3}{2}}}{32 a^{4}}+\frac {x^{4} \mathrm {arccosh}\left (a x \right )^{\frac {3}{2}}}{4}-\frac {3 \erf \left (\sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\right ) \sqrt {2}\, \sqrt {\pi }}{256 a^{4}}+\frac {3 \erfi \left (\sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\right ) \sqrt {2}\, \sqrt {\pi }}{256 a^{4}}-\frac {3 \erf \left (2 \sqrt {\mathrm {arccosh}\left (a x \right )}\right ) \sqrt {\pi }}{2048 a^{4}}+\frac {3 \erfi \left (2 \sqrt {\mathrm {arccosh}\left (a x \right )}\right ) \sqrt {\pi }}{2048 a^{4}}-\frac {9 x \sqrt {a x -1}\, \sqrt {a x +1}\, \sqrt {\mathrm {arccosh}\left (a x \right )}}{64 a^{3}}-\frac {3 x^{3} \sqrt {a x -1}\, \sqrt {a x +1}\, \sqrt {\mathrm {arccosh}\left (a x \right )}}{32 a} \]
command
int(x^3*arccosh(a*x)^(3/2),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| default | \(-\frac {\sqrt {2}\, \left (-32 \mathrm {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, a^{2} x^{2}+24 \sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\, a x +16 \mathrm {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }+3 \pi \erf \left (\sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\right )-3 \pi \erfi \left (\sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\right )\right )}{256 \sqrt {\pi }\, a^{4}}-\frac {-512 \mathrm {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, a^{4} x^{4}+192 \sqrt {\mathrm {arccosh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\, a^{3} x^{3}+512 \mathrm {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, a^{2} x^{2}-96 \sqrt {\mathrm {arccosh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\, a x -64 \mathrm {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {\pi }+3 \pi \erf \left (2 \sqrt {\mathrm {arccosh}\left (a x \right )}\right )-3 \pi \erfi \left (2 \sqrt {\mathrm {arccosh}\left (a x \right )}\right )}{2048 \sqrt {\pi }\, a^{4}}\) | \(242\) |
Maple 2021.1 output
\[ \int x^{3} \mathrm {arccosh}\left (a x \right )^{\frac {3}{2}}\, dx \]________________________________________________________________________________________