\[ \int x^3 \sinh ^{-1}(a x)^4 \, dx \]
Optimal antiderivative \[ -\frac {45 x^{2}}{128 a^{2}}+\frac {3 x^{4}}{128}-\frac {45 \arcsinh \! \left (a x \right )^{2}}{128 a^{4}}-\frac {9 x^{2} \arcsinh \! \left (a x \right )^{2}}{16 a^{2}}+\frac {3 x^{4} \arcsinh \! \left (a x \right )^{2}}{16}-\frac {3 \arcsinh \! \left (a x \right )^{4}}{32 a^{4}}+\frac {x^{4} \arcsinh \! \left (a x \right )^{4}}{4}+\frac {45 x \arcsinh \! \left (a x \right ) \sqrt {a^{2} x^{2}+1}}{64 a^{3}}-\frac {3 x^{3} \arcsinh \! \left (a x \right ) \sqrt {a^{2} x^{2}+1}}{32 a}+\frac {3 x \arcsinh \! \left (a x \right )^{3} \sqrt {a^{2} x^{2}+1}}{8 a^{3}}-\frac {x^{3} \arcsinh \! \left (a x \right )^{3} \sqrt {a^{2} x^{2}+1}}{4 a} \]
command
int(x^3*arcsinh(a*x)^4,x)
Maple 2022.1 output
\[\int x^{3} \arcsinh \left (a x \right )^{4}\, dx\]
Maple 2021.1 output
\[ \frac {\frac {a^{4} x^{4} \arcsinh \left (a x \right )^{4}}{4}-\frac {a^{3} x^{3} \arcsinh \left (a x \right )^{3} \sqrt {a^{2} x^{2}+1}}{4}+\frac {3 a x \arcsinh \left (a x \right )^{3} \sqrt {a^{2} x^{2}+1}}{8}-\frac {3 \arcsinh \left (a x \right )^{4}}{32}+\frac {3 a^{4} x^{4} \arcsinh \left (a x \right )^{2}}{16}-\frac {3 a^{3} x^{3} \arcsinh \left (a x \right ) \sqrt {a^{2} x^{2}+1}}{32}+\frac {45 \arcsinh \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a x}{64}+\frac {27 \arcsinh \left (a x \right )^{2}}{128}+\frac {3 a^{4} x^{4}}{128}-\frac {45 a^{2} x^{2}}{128}-\frac {45}{128}-\frac {9 \left (a^{2} x^{2}+1\right ) \arcsinh \left (a x \right )^{2}}{16}}{a^{4}} \]