\[ \int \frac {x^4 \sinh ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx \]
Optimal antiderivative \[ \frac {15 \arcsinh \! \left (a x \right )}{64 a^{5}}+\frac {3 x^{2} \arcsinh \! \left (a x \right )}{8 a^{3}}-\frac {x^{4} \arcsinh \! \left (a x \right )}{8 a}+\frac {\arcsinh \! \left (a x \right )^{3}}{8 a^{5}}-\frac {15 x \sqrt {a^{2} x^{2}+1}}{64 a^{4}}+\frac {x^{3} \sqrt {a^{2} x^{2}+1}}{32 a^{2}}-\frac {3 x \arcsinh \! \left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}}{8 a^{4}}+\frac {x^{3} \arcsinh \! \left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}}{4 a^{2}} \]
command
int(x^4*arcsinh(a*x)^2/(a^2*x^2+1)^(1/2),x)
Maple 2022.1 output
\[\int \frac {x^{4} \arcsinh \left (a x \right )^{2}}{\sqrt {a^{2} x^{2}+1}}\, dx\]
Maple 2021.1 output
\[ \frac {16 \arcsinh \left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}\, a^{3} x^{3}-8 \arcsinh \left (a x \right ) x^{4} a^{4}+2 \sqrt {a^{2} x^{2}+1}\, x^{3} a^{3}-24 \arcsinh \left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}\, a x +24 \arcsinh \left (a x \right ) x^{2} a^{2}+8 \arcsinh \left (a x \right )^{3}-15 \sqrt {a^{2} x^{2}+1}\, x a +15 \arcsinh \left (a x \right )}{64 a^{5}} \]