\[ \int \frac {x^4 \sinh ^{-1}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx \]
Optimal antiderivative \[ \frac {45 x^{2}}{128 a^{3}}-\frac {3 x^{4}}{128 a}+\frac {45 \arcsinh \! \left (a x \right )^{2}}{128 a^{5}}+\frac {9 x^{2} \arcsinh \! \left (a x \right )^{2}}{16 a^{3}}-\frac {3 x^{4} \arcsinh \! \left (a x \right )^{2}}{16 a}+\frac {3 \arcsinh \! \left (a x \right )^{4}}{32 a^{5}}-\frac {45 x \arcsinh \! \left (a x \right ) \sqrt {a^{2} x^{2}+1}}{64 a^{4}}+\frac {3 x^{3} \arcsinh \! \left (a x \right ) \sqrt {a^{2} x^{2}+1}}{32 a^{2}}-\frac {3 x \arcsinh \! \left (a x \right )^{3} \sqrt {a^{2} x^{2}+1}}{8 a^{4}}+\frac {x^{3} \arcsinh \! \left (a x \right )^{3} \sqrt {a^{2} x^{2}+1}}{4 a^{2}} \]
command
int(x^4*arcsinh(a*x)^3/(a^2*x^2+1)^(1/2),x)
Maple 2022.1 output
\[\int \frac {x^{4} \arcsinh \left (a x \right )^{3}}{\sqrt {a^{2} x^{2}+1}}\, dx\]
Maple 2021.1 output
\[ \frac {32 \arcsinh \left (a x \right )^{3} \sqrt {a^{2} x^{2}+1}\, a^{3} x^{3}-24 \arcsinh \left (a x \right )^{2} x^{4} a^{4}+12 \arcsinh \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a^{3} x^{3}-3 x^{4} a^{4}-48 \arcsinh \left (a x \right )^{3} \sqrt {a^{2} x^{2}+1}\, a x +72 \arcsinh \left (a x \right )^{2} a^{2} x^{2}+12 \arcsinh \left (a x \right )^{4}-90 \arcsinh \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a x +45 a^{2} x^{2}+45 \arcsinh \left (a x \right )^{2}+45}{128 a^{5}} \]