\[ \int \left (c+a^2 c x^2\right ) \sinh ^{-1}(a x)^3 \, dx \]
Optimal antiderivative \[ -\frac {2 c \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}{27 a}+\frac {14 c x \arcsinh \! \left (a x \right )}{3}+\frac {2 a^{2} c \,x^{3} \arcsinh \! \left (a x \right )}{9}-\frac {c \left (a^{2} x^{2}+1\right )^{\frac {3}{2}} \arcsinh \! \left (a x \right )^{2}}{3 a}+\frac {2 c x \arcsinh \! \left (a x \right )^{3}}{3}+\frac {c x \left (a^{2} x^{2}+1\right ) \arcsinh \! \left (a x \right )^{3}}{3}-\frac {40 c \sqrt {a^{2} x^{2}+1}}{9 a}-\frac {2 c \arcsinh \! \left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}}{a} \]
command
int((a^2*c*x^2+c)*arcsinh(a*x)^3,x)
Maple 2022.1 output
\[\int \left (a^{2} c \,x^{2}+c \right ) \arcsinh \left (a x \right )^{3}\, dx\]
Maple 2021.1 output
\[ \frac {c \left (9 \arcsinh \left (a x \right )^{3} a^{3} x^{3}-9 \arcsinh \left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}\, a^{2} x^{2}+27 \arcsinh \left (a x \right )^{3} a x +6 \arcsinh \left (a x \right ) a^{3} x^{3}-63 \arcsinh \left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}-2 \sqrt {a^{2} x^{2}+1}\, x^{2} a^{2}+126 a x \arcsinh \left (a x \right )-122 \sqrt {a^{2} x^{2}+1}\right )}{27 a} \]