\[ \int \frac {x^3 \sinh ^{-1}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx \]
Optimal antiderivative \[ \frac {40 x}{9 a^{3}}-\frac {2 x^{3}}{27 a}+\frac {2 x \arcsinh \! \left (a x \right )^{2}}{a^{3}}-\frac {x^{3} \arcsinh \! \left (a x \right )^{2}}{3 a}-\frac {40 \arcsinh \! \left (a x \right ) \sqrt {a^{2} x^{2}+1}}{9 a^{4}}+\frac {2 x^{2} \arcsinh \! \left (a x \right ) \sqrt {a^{2} x^{2}+1}}{9 a^{2}}-\frac {2 \arcsinh \! \left (a x \right )^{3} \sqrt {a^{2} x^{2}+1}}{3 a^{4}}+\frac {x^{2} \arcsinh \! \left (a x \right )^{3} \sqrt {a^{2} x^{2}+1}}{3 a^{2}} \]
command
int(x^3*arcsinh(a*x)^3/(a^2*x^2+1)^(1/2),x)
Maple 2022.1 output
\[\int \frac {x^{3} \arcsinh \left (a x \right )^{3}}{\sqrt {a^{2} x^{2}+1}}\, dx\]
Maple 2021.1 output
\[ \frac {9 \arcsinh \left (a x \right )^{3} x^{4} a^{4}-9 \arcsinh \left (a x \right )^{3} x^{2} a^{2}-9 \arcsinh \left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}\, a^{3} x^{3}+6 \arcsinh \left (a x \right ) x^{4} a^{4}-114 \arcsinh \left (a x \right ) x^{2} a^{2}-2 \sqrt {a^{2} x^{2}+1}\, x^{3} a^{3}-18 \arcsinh \left (a x \right )^{3}+54 \arcsinh \left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}\, a x -120 \arcsinh \left (a x \right )+120 \sqrt {a^{2} x^{2}+1}\, x a}{27 a^{4} \sqrt {a^{2} x^{2}+1}} \]