\[ \int \frac {x^3}{\cosh ^{-1}(a x)^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \erf \left (2 \sqrt {\mathrm {arccosh}\left (a x \right )}\right ) \sqrt {\pi }}{3 a^{4}}+\frac {2 \erfi \left (2 \sqrt {\mathrm {arccosh}\left (a x \right )}\right ) \sqrt {\pi }}{3 a^{4}}-\frac {\erf \left (\sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\right ) \sqrt {2}\, \sqrt {\pi }}{3 a^{4}}+\frac {\erfi \left (\sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\right ) \sqrt {2}\, \sqrt {\pi }}{3 a^{4}}-\frac {2 x^{3} \sqrt {a x -1}\, \sqrt {a x +1}}{3 a \mathrm {arccosh}\left (a x \right )^{\frac {3}{2}}}+\frac {4 x^{2}}{a^{2} \sqrt {\mathrm {arccosh}\left (a x \right )}}-\frac {16 x^{4}}{3 \sqrt {\mathrm {arccosh}\left (a x \right )}} \]
command
int(x^3/arccosh(a*x)^(5/2),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| default | \(-\frac {\sqrt {2}\, \left (4 \mathrm {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, a^{2} x^{2}+\sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\, a x +2 \mathrm {arccosh}\left (a x \right )^{2} \pi \erf \left (\sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\right )-2 \mathrm {arccosh}\left (a x \right )^{2} \pi \erfi \left (\sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\right )-2 \mathrm {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\right )}{6 \sqrt {\pi }\, a^{4} \mathrm {arccosh}\left (a x \right )^{2}}-\frac {16 \mathrm {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, a^{4} x^{4}+2 \sqrt {\mathrm {arccosh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\, a^{3} x^{3}-16 \mathrm {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, a^{2} x^{2}-\sqrt {\mathrm {arccosh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\, a x +2 \mathrm {arccosh}\left (a x \right )^{2} \pi \erf \left (2 \sqrt {\mathrm {arccosh}\left (a x \right )}\right )-2 \mathrm {arccosh}\left (a x \right )^{2} \pi \erfi \left (2 \sqrt {\mathrm {arccosh}\left (a x \right )}\right )+2 \sqrt {\pi }\, \mathrm {arccosh}\left (a x \right )^{\frac {3}{2}}}{3 \sqrt {\pi }\, a^{4} \mathrm {arccosh}\left (a x \right )^{2}}\) | \(277\) |
Maple 2021.1 output
\[ \int \frac {x^{3}}{\mathrm {arccosh}\left (a x \right )^{\frac {5}{2}}}\, dx \]________________________________________________________________________________________