\[ \int \left (c-a^2 c x^2\right )^3 \cosh ^{-1}(a x)^3 \, dx \]
Optimal antiderivative \[ \frac {4322 c^{3} x \,\mathrm {arccosh}\! \left (a x \right )}{1225}-\frac {1514 a^{2} c^{3} x^{3} \mathrm {arccosh}\! \left (a x \right )}{3675}+\frac {702 a^{4} c^{3} x^{5} \mathrm {arccosh}\! \left (a x \right )}{6125}-\frac {6 a^{6} c^{3} x^{7} \mathrm {arccosh}\! \left (a x \right )}{343}+\frac {8 c^{3} \left (a x -1\right )^{\frac {3}{2}} \left (a x +1\right )^{\frac {3}{2}} \mathrm {arccosh}\! \left (a x \right )^{2}}{35 a}-\frac {18 c^{3} \left (a x -1\right )^{\frac {5}{2}} \left (a x +1\right )^{\frac {5}{2}} \mathrm {arccosh}\! \left (a x \right )^{2}}{175 a}+\frac {3 c^{3} \left (a x -1\right )^{\frac {7}{2}} \left (a x +1\right )^{\frac {7}{2}} \mathrm {arccosh}\! \left (a x \right )^{2}}{49 a}+\frac {16 c^{3} x \mathrm {arccosh}\! \left (a x \right )^{3}}{35}+\frac {8 c^{3} x \left (-a^{2} x^{2}+1\right ) \mathrm {arccosh}\! \left (a x \right )^{3}}{35}+\frac {6 c^{3} x \left (-a^{2} x^{2}+1\right )^{2} \mathrm {arccosh}\! \left (a x \right )^{3}}{35}+\frac {c^{3} x \left (-a^{2} x^{2}+1\right )^{3} \mathrm {arccosh}\! \left (a x \right )^{3}}{7}+\frac {7104 c^{3} \left (-a^{2} x^{2}+1\right )}{42875 a \sqrt {a x -1}\, \sqrt {a x +1}}+\frac {1184 c^{3} \left (-a^{2} x^{2}+1\right )^{2}}{42875 a \sqrt {a x -1}\, \sqrt {a x +1}}+\frac {2664 c^{3} \left (-a^{2} x^{2}+1\right )^{3}}{214375 a \sqrt {a x -1}\, \sqrt {a x +1}}+\frac {6 c^{3} \left (-a^{2} x^{2}+1\right )^{4}}{2401 a \sqrt {a x -1}\, \sqrt {a x +1}}-\frac {976 c^{3} \sqrt {a x -1}\, \sqrt {a x +1}}{315 a}+\frac {16 a \,c^{3} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{315}-\frac {48 c^{3} \mathrm {arccosh}\! \left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{35 a} \]
command
int((-a^2*c*x^2+c)^3*arccosh(a*x)^3,x)
Maple 2022.1 output
\[\int \left (-a^{2} c \,x^{2}+c \right )^{3} \mathrm {arccosh}\left (a x \right )^{3}\, dx\]
Maple 2021.1 output
\[ -\frac {c^{3} \left (1929375 \mathrm {arccosh}\left (a x \right )^{3} a^{7} x^{7}-826875 \mathrm {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}\, a^{6} x^{6}-8103375 \mathrm {arccosh}\left (a x \right )^{3} a^{5} x^{5}+3869775 \mathrm {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}\, a^{4} x^{4}+236250 \,\mathrm {arccosh}\left (a x \right ) a^{7} x^{7}-33750 \sqrt {a x -1}\, \sqrt {a x +1}\, a^{6} x^{6}+13505625 a^{3} x^{3} \mathrm {arccosh}\left (a x \right )^{3}-8345925 \mathrm {arccosh}\left (a x \right )^{2} a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-1547910 \,\mathrm {arccosh}\left (a x \right ) a^{5} x^{5}+269082 \sqrt {a x +1}\, \sqrt {a x -1}\, x^{4} a^{4}-13505625 a x \mathrm {arccosh}\left (a x \right )^{3}+23825025 \mathrm {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}+5563950 a^{3} x^{3} \mathrm {arccosh}\left (a x \right )-1495874 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-47650050 a x \,\mathrm {arccosh}\left (a x \right )+44658302 \sqrt {a x -1}\, \sqrt {a x +1}\right )}{13505625 a} \]