\[ \int \left (c+a^2 c x^2\right )^3 \sinh ^{-1}(a x)^3 \, dx \]
Optimal antiderivative \[ -\frac {30256 c^{3} \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}{385875 a}-\frac {2664 c^{3} \left (a^{2} x^{2}+1\right )^{\frac {5}{2}}}{214375 a}-\frac {6 c^{3} \left (a^{2} x^{2}+1\right )^{\frac {7}{2}}}{2401 a}+\frac {4322 c^{3} x \arcsinh \! \left (a x \right )}{1225}+\frac {1514 a^{2} c^{3} x^{3} \arcsinh \! \left (a x \right )}{3675}+\frac {702 a^{4} c^{3} x^{5} \arcsinh \! \left (a x \right )}{6125}+\frac {6 a^{6} c^{3} x^{7} \arcsinh \! \left (a x \right )}{343}-\frac {8 c^{3} \left (a^{2} x^{2}+1\right )^{\frac {3}{2}} \arcsinh \! \left (a x \right )^{2}}{35 a}-\frac {18 c^{3} \left (a^{2} x^{2}+1\right )^{\frac {5}{2}} \arcsinh \! \left (a x \right )^{2}}{175 a}-\frac {3 c^{3} \left (a^{2} x^{2}+1\right )^{\frac {7}{2}} \arcsinh \! \left (a x \right )^{2}}{49 a}+\frac {16 c^{3} x \arcsinh \! \left (a x \right )^{3}}{35}+\frac {8 c^{3} x \left (a^{2} x^{2}+1\right ) \arcsinh \! \left (a x \right )^{3}}{35}+\frac {6 c^{3} x \left (a^{2} x^{2}+1\right )^{2} \arcsinh \! \left (a x \right )^{3}}{35}+\frac {c^{3} x \left (a^{2} x^{2}+1\right )^{3} \arcsinh \! \left (a x \right )^{3}}{7}-\frac {413312 c^{3} \sqrt {a^{2} x^{2}+1}}{128625 a}-\frac {48 c^{3} \arcsinh \! \left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}}{35 a} \]
command
int((a^2*c*x^2+c)^3*arcsinh(a*x)^3,x)
Maple 2022.1 output
\[\int \left (a^{2} c \,x^{2}+c \right )^{3} \arcsinh \left (a x \right )^{3}\, dx\]
Maple 2021.1 output
\[ \frac {c^{3} \left (1929375 \arcsinh \left (a x \right )^{3} a^{7} x^{7}-826875 \arcsinh \left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}\, a^{6} x^{6}+8103375 \arcsinh \left (a x \right )^{3} a^{5} x^{5}+236250 \arcsinh \left (a x \right ) a^{7} x^{7}-3869775 \arcsinh \left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}\, a^{4} x^{4}-33750 a^{6} x^{6} \sqrt {a^{2} x^{2}+1}+13505625 \arcsinh \left (a x \right )^{3} a^{3} x^{3}+1547910 \arcsinh \left (a x \right ) a^{5} x^{5}-8345925 \arcsinh \left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}\, a^{2} x^{2}-269082 \sqrt {a^{2} x^{2}+1}\, x^{4} a^{4}+13505625 \arcsinh \left (a x \right )^{3} a x +5563950 \arcsinh \left (a x \right ) a^{3} x^{3}-23825025 \arcsinh \left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}-1495874 \sqrt {a^{2} x^{2}+1}\, x^{2} a^{2}+47650050 a x \arcsinh \left (a x \right )-44658302 \sqrt {a^{2} x^{2}+1}\right )}{13505625 a} \]