\[ \int \frac {x^3}{\cosh ^{-1}(a x)^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {\erf \left (\sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\right ) \sqrt {2}\, \sqrt {\pi }}{4 a^{4}}+\frac {\erfi \left (\sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\right ) \sqrt {2}\, \sqrt {\pi }}{4 a^{4}}+\frac {\erf \left (2 \sqrt {\mathrm {arccosh}\left (a x \right )}\right ) \sqrt {\pi }}{4 a^{4}}+\frac {\erfi \left (2 \sqrt {\mathrm {arccosh}\left (a x \right )}\right ) \sqrt {\pi }}{4 a^{4}}-\frac {2 x^{3} \sqrt {a x -1}\, \sqrt {a x +1}}{a \sqrt {\mathrm {arccosh}\left (a x \right )}} \]
command
int(x^3/arccosh(a*x)^(3/2),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| default | \(-\frac {\sqrt {2}\, \left (2 \sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\, a x -\mathrm {arccosh}\left (a x \right ) \pi \erf \left (\sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\right )-\mathrm {arccosh}\left (a x \right ) \pi \erfi \left (\sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\right )\right )}{4 \sqrt {\pi }\, a^{4} \mathrm {arccosh}\left (a x \right )}-\frac {8 \sqrt {\mathrm {arccosh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\, a^{3} x^{3}-4 \sqrt {\mathrm {arccosh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\, a x -\mathrm {arccosh}\left (a x \right ) \pi \erf \left (2 \sqrt {\mathrm {arccosh}\left (a x \right )}\right )-\mathrm {arccosh}\left (a x \right ) \pi \erfi \left (2 \sqrt {\mathrm {arccosh}\left (a x \right )}\right )}{4 \sqrt {\pi }\, a^{4} \mathrm {arccosh}\left (a x \right )}\) | \(191\) |
Maple 2021.1 output
\[ \int \frac {x^{3}}{\mathrm {arccosh}\left (a x \right )^{\frac {3}{2}}}\, dx \]________________________________________________________________________________________