11.2 Problem number 13

\[ \int x^3 \sinh ^{-1}(a x)^2 \, dx \]

Optimal antiderivative \[ -\frac {3 x^{2}}{32 a^{2}}+\frac {x^{4}}{32}-\frac {3 \arcsinh \! \left (a x \right )^{2}}{32 a^{4}}+\frac {x^{4} \arcsinh \! \left (a x \right )^{2}}{4}+\frac {3 x \arcsinh \! \left (a x \right ) \sqrt {a^{2} x^{2}+1}}{16 a^{3}}-\frac {x^{3} \arcsinh \! \left (a x \right ) \sqrt {a^{2} x^{2}+1}}{8 a} \]

command

int(x^3*arcsinh(a*x)^2,x)

Maple 2022.1 output

\[\int x^{3} \arcsinh \left (a x \right )^{2}\, dx\]

Maple 2021.1 output

\[ \frac {\frac {a^{4} x^{4} \arcsinh \left (a x \right )^{2}}{4}-\frac {a^{3} x^{3} \arcsinh \left (a x \right ) \sqrt {a^{2} x^{2}+1}}{8}+\frac {3 \arcsinh \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a x}{16}-\frac {3 \arcsinh \left (a x \right )^{2}}{32}+\frac {a^{4} x^{4}}{32}-\frac {3 a^{2} x^{2}}{32}-\frac {3}{32}}{a^{4}} \]