\[ \int x^3 \sinh ^{-1}(a x)^3 \, dx \]
Optimal antiderivative \[ -\frac {45 \arcsinh \! \left (a x \right )}{256 a^{4}}-\frac {9 x^{2} \arcsinh \! \left (a x \right )}{32 a^{2}}+\frac {3 x^{4} \arcsinh \! \left (a x \right )}{32}-\frac {3 \arcsinh \! \left (a x \right )^{3}}{32 a^{4}}+\frac {x^{4} \arcsinh \! \left (a x \right )^{3}}{4}+\frac {45 x \sqrt {a^{2} x^{2}+1}}{256 a^{3}}-\frac {3 x^{3} \sqrt {a^{2} x^{2}+1}}{128 a}+\frac {9 x \arcsinh \! \left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}}{32 a^{3}}-\frac {3 x^{3} \arcsinh \! \left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}}{16 a} \]
command
int(x^3*arcsinh(a*x)^3,x)
Maple 2022.1 output
\[\int x^{3} \arcsinh \left (a x \right )^{3}\, dx\]
Maple 2021.1 output
\[ \frac {\frac {a^{4} x^{4} \arcsinh \left (a x \right )^{3}}{4}-\frac {3 a^{3} x^{3} \arcsinh \left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}}{16}+\frac {9 a x \arcsinh \left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}}{32}-\frac {3 \arcsinh \left (a x \right )^{3}}{32}+\frac {3 \arcsinh \left (a x \right ) a^{4} x^{4}}{32}-\frac {3 a^{3} x^{3} \sqrt {a^{2} x^{2}+1}}{128}+\frac {45 a x \sqrt {a^{2} x^{2}+1}}{256}+\frac {27 \arcsinh \left (a x \right )}{256}-\frac {9 \left (a^{2} x^{2}+1\right ) \arcsinh \left (a x \right )}{32}}{a^{4}} \]