\[ \int x^4 \sinh ^{-1}(a x)^3 \, dx \]
Optimal antiderivative \[ \frac {76 \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}{1125 a^{5}}-\frac {6 \left (a^{2} x^{2}+1\right )^{\frac {5}{2}}}{625 a^{5}}+\frac {16 x \arcsinh \! \left (a x \right )}{25 a^{4}}-\frac {8 x^{3} \arcsinh \! \left (a x \right )}{75 a^{2}}+\frac {6 x^{5} \arcsinh \! \left (a x \right )}{125}+\frac {x^{5} \arcsinh \! \left (a x \right )^{3}}{5}-\frac {298 \sqrt {a^{2} x^{2}+1}}{375 a^{5}}-\frac {8 \arcsinh \! \left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}}{25 a^{5}}+\frac {4 x^{2} \arcsinh \! \left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}}{25 a^{3}}-\frac {3 x^{4} \arcsinh \! \left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}}{25 a} \]
command
int(x^4*arcsinh(a*x)^3,x)
Maple 2022.1 output
\[\int x^{4} \arcsinh \left (a x \right )^{3}\, dx\]
Maple 2021.1 output
\[ \frac {\frac {a^{5} x^{5} \arcsinh \left (a x \right )^{3}}{5}-\frac {8 \arcsinh \left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}}{25}-\frac {3 a^{4} x^{4} \arcsinh \left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}}{25}+\frac {4 \arcsinh \left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}\, a^{2} x^{2}}{25}+\frac {16 a x \arcsinh \left (a x \right )}{25}-\frac {4144 \sqrt {a^{2} x^{2}+1}}{5625}+\frac {6 a^{5} x^{5} \arcsinh \left (a x \right )}{125}-\frac {6 a^{4} x^{4} \sqrt {a^{2} x^{2}+1}}{625}+\frac {272 a^{2} x^{2} \sqrt {a^{2} x^{2}+1}}{5625}-\frac {8 a^{3} x^{3} \arcsinh \left (a x \right )}{75}}{a^{5}} \]