\[ \int \left (c-a^2 c x^2\right )^3 \cosh ^{-1}(a x)^2 \, dx \]
Optimal antiderivative \[ \frac {4322 c^{3} x}{3675}-\frac {1514 a^{2} c^{3} x^{3}}{11025}+\frac {234 a^{4} c^{3} x^{5}}{6125}-\frac {2 a^{6} c^{3} x^{7}}{343}+\frac {16 c^{3} \left (a x -1\right )^{\frac {3}{2}} \left (a x +1\right )^{\frac {3}{2}} \mathrm {arccosh}\! \left (a x \right )}{105 a}-\frac {12 c^{3} \left (a x -1\right )^{\frac {5}{2}} \left (a x +1\right )^{\frac {5}{2}} \mathrm {arccosh}\! \left (a x \right )}{175 a}+\frac {2 c^{3} \left (a x -1\right )^{\frac {7}{2}} \left (a x +1\right )^{\frac {7}{2}} \mathrm {arccosh}\! \left (a x \right )}{49 a}+\frac {16 c^{3} x \mathrm {arccosh}\! \left (a x \right )^{2}}{35}+\frac {8 c^{3} x \left (-a^{2} x^{2}+1\right ) \mathrm {arccosh}\! \left (a x \right )^{2}}{35}+\frac {6 c^{3} x \left (-a^{2} x^{2}+1\right )^{2} \mathrm {arccosh}\! \left (a x \right )^{2}}{35}+\frac {c^{3} x \left (-a^{2} x^{2}+1\right )^{3} \mathrm {arccosh}\! \left (a x \right )^{2}}{7}-\frac {32 c^{3} \mathrm {arccosh}\! \left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}}{35 a} \]
command
int((-a^2*c*x^2+c)^3*arccosh(a*x)^2,x)
Maple 2022.1 output
\[\int \left (-a^{2} c \,x^{2}+c \right )^{3} \mathrm {arccosh}\left (a x \right )^{2}\, dx\]
Maple 2021.1 output
\[ -\frac {c^{3} \left (55125 \mathrm {arccosh}\left (a x \right )^{2} a^{7} x^{7}-15750 \,\mathrm {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}\, a^{6} x^{6}-231525 \mathrm {arccosh}\left (a x \right )^{2} a^{5} x^{5}+73710 \,\mathrm {arccosh}\left (a x \right ) a^{4} x^{4} \sqrt {a x -1}\, \sqrt {a x +1}+2250 x^{7} a^{7}+385875 a^{3} x^{3} \mathrm {arccosh}\left (a x \right )^{2}-158970 \,\mathrm {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}\, a^{2} x^{2}-14742 x^{5} a^{5}-385875 a x \mathrm {arccosh}\left (a x \right )^{2}+453810 \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right )+52990 x^{3} a^{3}-453810 a x \right )}{385875 a} \]