\[ \int \left (c-a^2 c x^2\right )^2 \cosh ^{-1}(a x)^2 \, dx \]
Optimal antiderivative \[ \frac {298 c^{2} x}{225}-\frac {76 a^{2} c^{2} x^{3}}{675}+\frac {2 a^{4} c^{2} x^{5}}{125}+\frac {8 c^{2} \left (a x -1\right )^{\frac {3}{2}} \left (a x +1\right )^{\frac {3}{2}} \mathrm {arccosh}\! \left (a x \right )}{45 a}-\frac {2 c^{2} \left (a x -1\right )^{\frac {5}{2}} \left (a x +1\right )^{\frac {5}{2}} \mathrm {arccosh}\! \left (a x \right )}{25 a}+\frac {8 c^{2} x \mathrm {arccosh}\! \left (a x \right )^{2}}{15}+\frac {4 c^{2} x \left (-a^{2} x^{2}+1\right ) \mathrm {arccosh}\! \left (a x \right )^{2}}{15}+\frac {c^{2} x \left (-a^{2} x^{2}+1\right )^{2} \mathrm {arccosh}\! \left (a x \right )^{2}}{5}-\frac {16 c^{2} \mathrm {arccosh}\! \left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}}{15 a} \]
command
int((-a^2*c*x^2+c)^2*arccosh(a*x)^2,x)
Maple 2022.1 output
\[\int \left (-a^{2} c \,x^{2}+c \right )^{2} \mathrm {arccosh}\left (a x \right )^{2}\, dx\]
Maple 2021.1 output
\[ \frac {c^{2} \left (675 \mathrm {arccosh}\left (a x \right )^{2} a^{5} x^{5}-270 \,\mathrm {arccosh}\left (a x \right ) a^{4} x^{4} \sqrt {a x -1}\, \sqrt {a x +1}-2250 a^{3} x^{3} \mathrm {arccosh}\left (a x \right )^{2}+1140 \,\mathrm {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}\, a^{2} x^{2}+54 x^{5} a^{5}+3375 a x \mathrm {arccosh}\left (a x \right )^{2}-4470 \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right )-380 x^{3} a^{3}+4470 a x \right )}{3375 a} \]