\[ \int \frac {x^3 \sinh ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}{27 a^{4}}+\frac {4 x \arcsinh \! \left (a x \right )}{3 a^{3}}-\frac {2 x^{3} \arcsinh \! \left (a x \right )}{9 a}-\frac {14 \sqrt {a^{2} x^{2}+1}}{9 a^{4}}-\frac {2 \arcsinh \! \left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}}{3 a^{4}}+\frac {x^{2} \arcsinh \! \left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}}{3 a^{2}} \]
command
int(x^3*arcsinh(a*x)^2/(a^2*x^2+1)^(1/2),x)
Maple 2022.1 output
\[\int \frac {x^{3} \arcsinh \left (a x \right )^{2}}{\sqrt {a^{2} x^{2}+1}}\, dx\]
Maple 2021.1 output
\[ \frac {9 \arcsinh \left (a x \right )^{2} x^{4} a^{4}-9 \arcsinh \left (a x \right )^{2} a^{2} x^{2}-6 \arcsinh \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a^{3} x^{3}+2 x^{4} a^{4}-38 a^{2} x^{2}-18 \arcsinh \left (a x \right )^{2}+36 \arcsinh \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a x -40}{27 a^{4} \sqrt {a^{2} x^{2}+1}} \]