\[ \int \frac {x^3}{\cosh ^{-1}(a x)^{7/2}} \, dx \]
Optimal antiderivative \[ \frac {4 x^{2}}{5 a^{2} \mathrm {arccosh}\left (a x \right )^{\frac {3}{2}}}-\frac {16 x^{4}}{15 \mathrm {arccosh}\left (a x \right )^{\frac {3}{2}}}+\frac {16 \erf \left (2 \sqrt {\mathrm {arccosh}\left (a x \right )}\right ) \sqrt {\pi }}{15 a^{4}}+\frac {16 \erfi \left (2 \sqrt {\mathrm {arccosh}\left (a x \right )}\right ) \sqrt {\pi }}{15 a^{4}}+\frac {4 \erf \left (\sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\right ) \sqrt {2}\, \sqrt {\pi }}{15 a^{4}}+\frac {4 \erfi \left (\sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\right ) \sqrt {2}\, \sqrt {\pi }}{15 a^{4}}-\frac {2 x^{3} \sqrt {a x -1}\, \sqrt {a x +1}}{5 a \mathrm {arccosh}\left (a x \right )^{\frac {5}{2}}}+\frac {16 x \sqrt {a x -1}\, \sqrt {a x +1}}{5 a^{3} \sqrt {\mathrm {arccosh}\left (a x \right )}}-\frac {128 x^{3} \sqrt {a x -1}\, \sqrt {a x +1}}{15 a \sqrt {\mathrm {arccosh}\left (a x \right )}} \]
command
int(x^3/arccosh(a*x)^(7/2),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| default | \(\frac {\sqrt {2}\, \left (-16 \mathrm {arccosh}\left (a x \right )^{\frac {5}{2}} \sqrt {2}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\, a x -4 \mathrm {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, a^{2} x^{2}-3 \sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\, a x +8 \mathrm {arccosh}\left (a x \right )^{3} \pi \erf \left (\sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\right )+8 \mathrm {arccosh}\left (a x \right )^{3} \pi \erfi \left (\sqrt {2}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\right )+2 \mathrm {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\right )}{30 \sqrt {\pi }\, a^{4} \mathrm {arccosh}\left (a x \right )^{3}}+\frac {-128 \sqrt {a x -1}\, \sqrt {a x +1}\, \sqrt {\pi }\, \mathrm {arccosh}\left (a x \right )^{\frac {5}{2}} a^{3} x^{3}-16 \mathrm {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, a^{4} x^{4}-6 \sqrt {\mathrm {arccosh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\, a^{3} x^{3}+64 \sqrt {a x -1}\, \sqrt {a x +1}\, \sqrt {\pi }\, \mathrm {arccosh}\left (a x \right )^{\frac {5}{2}} a x +16 \mathrm {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, a^{2} x^{2}+3 \sqrt {\mathrm {arccosh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\, a x +16 \mathrm {arccosh}\left (a x \right )^{3} \pi \erf \left (2 \sqrt {\mathrm {arccosh}\left (a x \right )}\right )+16 \mathrm {arccosh}\left (a x \right )^{3} \pi \erfi \left (2 \sqrt {\mathrm {arccosh}\left (a x \right )}\right )-2 \sqrt {\pi }\, \mathrm {arccosh}\left (a x \right )^{\frac {3}{2}}}{15 \sqrt {\pi }\, a^{4} \mathrm {arccosh}\left (a x \right )^{3}}\) | \(366\) |
Maple 2021.1 output
\[ \int \frac {x^{3}}{\mathrm {arccosh}\left (a x \right )^{\frac {7}{2}}}\, dx \]________________________________________________________________________________________