\[ \int \left (c-a^2 c x^2\right )^2 \cosh ^{-1}(a x)^3 \, dx \]
Optimal antiderivative \[ \frac {298 c^{2} x \,\mathrm {arccosh}\! \left (a x \right )}{75}-\frac {76 a^{2} c^{2} x^{3} \mathrm {arccosh}\! \left (a x \right )}{225}+\frac {6 a^{4} c^{2} x^{5} \mathrm {arccosh}\! \left (a x \right )}{125}+\frac {4 c^{2} \left (a x -1\right )^{\frac {3}{2}} \left (a x +1\right )^{\frac {3}{2}} \mathrm {arccosh}\! \left (a x \right )^{2}}{15 a}-\frac {3 c^{2} \left (a x -1\right )^{\frac {5}{2}} \left (a x +1\right )^{\frac {5}{2}} \mathrm {arccosh}\! \left (a x \right )^{2}}{25 a}+\frac {8 c^{2} x \mathrm {arccosh}\! \left (a x \right )^{3}}{15}+\frac {4 c^{2} x \left (-a^{2} x^{2}+1\right ) \mathrm {arccosh}\! \left (a x \right )^{3}}{15}+\frac {c^{2} x \left (-a^{2} x^{2}+1\right )^{2} \mathrm {arccosh}\! \left (a x \right )^{3}}{5}+\frac {16 c^{2} \left (-a^{2} x^{2}+1\right )}{125 a \sqrt {a x -1}\, \sqrt {a x +1}}+\frac {8 c^{2} \left (-a^{2} x^{2}+1\right )^{2}}{375 a \sqrt {a x -1}\, \sqrt {a x +1}}+\frac {6 c^{2} \left (-a^{2} x^{2}+1\right )^{3}}{625 a \sqrt {a x -1}\, \sqrt {a x +1}}-\frac {488 c^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{135 a}+\frac {8 a \,c^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{135}-\frac {8 c^{2} \mathrm {arccosh}\! \left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{5 a} \]
command
int((-a^2*c*x^2+c)^2*arccosh(a*x)^3,x)
Maple 2022.1 output
\[\int \left (-a^{2} c \,x^{2}+c \right )^{2} \mathrm {arccosh}\left (a x \right )^{3}\, dx\]
Maple 2021.1 output
\[ \frac {c^{2} \left (3375 \mathrm {arccosh}\left (a x \right )^{3} a^{5} x^{5}-2025 \mathrm {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}\, a^{4} x^{4}-11250 a^{3} x^{3} \mathrm {arccosh}\left (a x \right )^{3}+8550 \mathrm {arccosh}\left (a x \right )^{2} a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}+810 \,\mathrm {arccosh}\left (a x \right ) a^{5} x^{5}-162 \sqrt {a x +1}\, \sqrt {a x -1}\, x^{4} a^{4}+16875 a x \mathrm {arccosh}\left (a x \right )^{3}-33525 \mathrm {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}-5700 a^{3} x^{3} \mathrm {arccosh}\left (a x \right )+1684 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}+67050 a x \,\mathrm {arccosh}\left (a x \right )-63682 \sqrt {a x -1}\, \sqrt {a x +1}\right )}{16875 a} \]