15.3 Problem number 240

$$\int {(c-a^{2}c{x}^2)}^3{\cosh }^{-1}(ax{)}^{3}dx$$

Optimal antiderivative $$\frac{4322c^{3}x\mathrm{arccosh}\left(ax\right)}{1225}-\frac{1514a^2{c}^3{x}^3\mathrm{arccosh}\left(ax\right)}{3675}+\frac{702a^4{c}^3{x}^5\mathrm{arccosh}\left(ax\right)}{6125}-\frac{6a^6{c}^3{x}^7\mathrm{arccosh}\left(ax\right)}{343}+\frac{8c^3{(ax-1)}^{\frac{3}{2}}{(ax+1)}^{\frac{3}{2}}\mathrm{arccosh}{\left(ax\right)}^2}{35a}-\frac{18c^3{(ax-1)}^{\frac{5}{2}}{(ax+1)}^{\frac{5}{2}}\mathrm{arccosh}{\left(ax\right)}^2}{175a}+\frac{3c^3{(ax-1)}^{\frac{7}{2}}{(ax+1)}^{\frac{7}{2}}\mathrm{arccosh}{\left(ax\right)}^2}{49a}+\frac{16c^{3}x\mathrm{arccosh}{\left(ax\right)}^3}{35}+\frac{8c^{3}x(-a^2{x}^2+1)\mathrm{arccosh}{\left(ax\right)}^3}{35}+\frac{6c^{3}x{(-a^2{x}^2+1)}^2\mathrm{arccosh}{\left(ax\right)}^3}{35}+\frac{c^{3}x{(-a^2{x}^2+1)}^3\mathrm{arccosh}{\left(ax\right)}^3}{7}+\frac{7104c^3(-a^2{x}^2+1)}{42875a\sqrt{ax-1}\sqrt{ax+1}}+\frac{1184c^3{(-a^2{x}^2+1)}^2}{42875a\sqrt{ax-1}\sqrt{ax+1}}+\frac{2664c^3{(-a^2{x}^2+1)}^3}{214375a\sqrt{ax-1}\sqrt{ax+1}}+\frac{6c^3{(-a^2{x}^2+1)}^4}{2401a\sqrt{ax-1}\sqrt{ax+1}}-\frac{976c^3\sqrt{ax-1}\sqrt{ax+1}}{315a}+\frac{16a{c}^3{x}^2\sqrt{ax-1}\sqrt{ax+1}}{315}-\frac{48c^3\mathrm{arccosh}{\left(ax\right)}^2\sqrt{ax-1}\sqrt{ax+1}}{35a}$$

command

int((-a^2*c*x^2+c)^3*arccosh(a*x)^3,x)

Maple 2022.1 output

$$\int {(-a^{2}c{x}^2+c)}^3\mathrm{arccosh}{\left(ax\right)}^{3}dx$$

Maple 2021.1 output

$$-\frac{c^3(1929375\mathrm{arccosh}{\left(ax\right)}^3{a}^7{x}^7-826875\mathrm{arccosh}{\left(ax\right)}^2\sqrt{ax-1}\sqrt{ax+1}{a}^6{x}^6-8103375\mathrm{arccosh}{\left(ax\right)}^3{a}^5{x}^5+3869775\mathrm{arccosh}{\left(ax\right)}^2\sqrt{ax-1}\sqrt{ax+1}{a}^4{x}^4+236250\mathrm{arccosh}\left(ax\right)a^7{x}^7-33750\sqrt{ax-1}\sqrt{ax+1}{a}^6{x}^6+13505625a^3{x}^3\mathrm{arccosh}{\left(ax\right)}^3-8345925\mathrm{arccosh}{\left(ax\right)}^2{a}^2{x}^2\sqrt{ax-1}\sqrt{ax+1}-1547910\mathrm{arccosh}\left(ax\right)a^5{x}^5+269082\sqrt{ax+1}\sqrt{ax-1}{x}^4{a}^4-13505625ax\mathrm{arccosh}{\left(ax\right)}^3+23825025\mathrm{arccosh}{\left(ax\right)}^2\sqrt{ax-1}\sqrt{ax+1}+5563950a^3{x}^3\mathrm{arccosh}\left(ax\right)-1495874a^2{x}^2\sqrt{ax-1}\sqrt{ax+1}-47650050ax\mathrm{arccosh}\left(ax\right)[phone]\mathrm{}\sqrt{ax-1}\sqrt{ax+1})}{13505625a}$$