\[ \int x^3 \sqrt {\cosh ^{-1}(a x)} \, dx \]
Optimal antiderivative \[ -\frac {\erf \left (\sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\right ) \sqrt {2}\, \sqrt {\pi }}{64 a^{4}}-\frac {\erfi \left (\sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\right ) \sqrt {2}\, \sqrt {\pi }}{64 a^{4}}-\frac {\erf \left (2 \sqrt {\mathrm {arccosh}\left (a x \right )}\right ) \sqrt {\pi }}{256 a^{4}}-\frac {\erfi \left (2 \sqrt {\mathrm {arccosh}\left (a x \right )}\right ) \sqrt {\pi }}{256 a^{4}}-\frac {3 \sqrt {\mathrm {arccosh}\left (a x \right )}}{32 a^{4}}+\frac {x^{4} \sqrt {\mathrm {arccosh}\left (a x \right )}}{4} \]
command
int(x^3*arccosh(a*x)^(1/2),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| default | \(-\frac {\sqrt {2}\, \left (-8 \sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\, \sqrt {\pi }\, a^{2} x^{2}+4 \sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\, \sqrt {\pi }+\pi \erf \left (\sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\right )+\pi \erfi \left (\sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\right )\right )}{64 \sqrt {\pi }\, a^{4}}-\frac {-64 \sqrt {\mathrm {arccosh}\left (a x \right )}\, \sqrt {\pi }\, a^{4} x^{4}+64 \sqrt {\mathrm {arccosh}\left (a x \right )}\, \sqrt {\pi }\, a^{2} x^{2}+\pi \erf \left (2 \sqrt {\mathrm {arccosh}\left (a x \right )}\right )+\pi \erfi \left (2 \sqrt {\mathrm {arccosh}\left (a x \right )}\right )-8 \sqrt {\mathrm {arccosh}\left (a x \right )}\, \sqrt {\pi }}{256 \sqrt {\pi }\, a^{4}}\) | \(150\) |
Maple 2021.1 output
\[ \int x^{3} \sqrt {\mathrm {arccosh}\left (a x \right )}\, dx \]________________________________________________________________________________________