\[ \int \frac {x^3}{\sqrt {\cosh ^{-1}(a x)}} \, dx \]
Optimal antiderivative \[ -\frac {\erf \left (\sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\right ) \sqrt {2}\, \sqrt {\pi }}{16 a^{4}}+\frac {\erfi \left (\sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\right ) \sqrt {2}\, \sqrt {\pi }}{16 a^{4}}-\frac {\erf \left (2 \sqrt {\mathrm {arccosh}\left (a x \right )}\right ) \sqrt {\pi }}{32 a^{4}}+\frac {\erfi \left (2 \sqrt {\mathrm {arccosh}\left (a x \right )}\right ) \sqrt {\pi }}{32 a^{4}} \]
command
int(x^3/arccosh(a*x)^(1/2),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| default | \(-\frac {\sqrt {\pi }\, \sqrt {2}\, \left (\erf \left (\sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\right )-\erfi \left (\sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\right )\right )}{16 a^{4}}-\frac {\sqrt {\pi }\, \left (\erf \left (2 \sqrt {\mathrm {arccosh}\left (a x \right )}\right )-\erfi \left (2 \sqrt {\mathrm {arccosh}\left (a x \right )}\right )\right )}{32 a^{4}}\) | \(67\) |
Maple 2021.1 output
\[ \int \frac {x^{3}}{\sqrt {\mathrm {arccosh}\left (a x \right )}}\, dx \]________________________________________________________________________________________