\[ \int x^3 \cosh ^{-1}(a x)^{5/2} \, dx \]
Optimal antiderivative \[ -\frac {3 \mathrm {arccosh}\left (a x \right )^{\frac {5}{2}}}{32 a^{4}}+\frac {x^{4} \mathrm {arccosh}\left (a x \right )^{\frac {5}{2}}}{4}-\frac {15 \erf \left (\sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\right ) \sqrt {2}\, \sqrt {\pi }}{1024 a^{4}}-\frac {15 \erfi \left (\sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\right ) \sqrt {2}\, \sqrt {\pi }}{1024 a^{4}}-\frac {15 \erf \left (2 \sqrt {\mathrm {arccosh}\left (a x \right )}\right ) \sqrt {\pi }}{16384 a^{4}}-\frac {15 \erfi \left (2 \sqrt {\mathrm {arccosh}\left (a x \right )}\right ) \sqrt {\pi }}{16384 a^{4}}-\frac {15 x \mathrm {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {a x -1}\, \sqrt {a x +1}}{64 a^{3}}-\frac {5 x^{3} \mathrm {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {a x -1}\, \sqrt {a x +1}}{32 a}-\frac {225 \sqrt {\mathrm {arccosh}\left (a x \right )}}{2048 a^{4}}+\frac {45 x^{2} \sqrt {\mathrm {arccosh}\left (a x \right )}}{256 a^{2}}+\frac {15 x^{4} \sqrt {\mathrm {arccosh}\left (a x \right )}}{256} \]
command
int(x^3*arccosh(a*x)^(5/2),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| default | \(-\frac {\sqrt {2}\, \left (-128 \mathrm {arccosh}\left (a x \right )^{\frac {5}{2}} \sqrt {2}\, \sqrt {\pi }\, a^{2} x^{2}+160 \mathrm {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\, a x -120 \sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\, \sqrt {\pi }\, a^{2} x^{2}+64 \mathrm {arccosh}\left (a x \right )^{\frac {5}{2}} \sqrt {2}\, \sqrt {\pi }+60 \sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\, \sqrt {\pi }+15 \pi \erf \left (\sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\right )+15 \pi \erfi \left (\sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\right )\right )}{1024 \sqrt {\pi }\, a^{4}}-\frac {-4096 \mathrm {arccosh}\left (a x \right )^{\frac {5}{2}} \sqrt {\pi }\, a^{4} x^{4}+2560 \mathrm {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\, a^{3} x^{3}-960 \sqrt {\mathrm {arccosh}\left (a x \right )}\, \sqrt {\pi }\, a^{4} x^{4}+4096 \mathrm {arccosh}\left (a x \right )^{\frac {5}{2}} \sqrt {\pi }\, a^{2} x^{2}-1280 \mathrm {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\, a x +960 \sqrt {\mathrm {arccosh}\left (a x \right )}\, \sqrt {\pi }\, a^{2} x^{2}-512 \mathrm {arccosh}\left (a x \right )^{\frac {5}{2}} \sqrt {\pi }-120 \sqrt {\mathrm {arccosh}\left (a x \right )}\, \sqrt {\pi }+15 \pi \erf \left (2 \sqrt {\mathrm {arccosh}\left (a x \right )}\right )+15 \pi \erfi \left (2 \sqrt {\mathrm {arccosh}\left (a x \right )}\right )}{16384 \sqrt {\pi }\, a^{4}}\) | \(321\) |
Maple 2021.1 output
\[ \int x^{3} \mathrm {arccosh}\left (a x \right )^{\frac {5}{2}}\, dx \]________________________________________________________________________________________