16.1 Problem number 8

$$\int (d+ex{)}^3{\cosh }^{-1}(cx{)}^{2}dx$$

Optimal antiderivative $$2d^{3}x+\frac{4d{e}^{2}x}{3c^2}+\frac{3d^{2}e{x}^2}{4}+\frac{3e^3{x}^2}{32c^2}+\frac{2d{e}^2{x}^3}{9}+\frac{e^3{x}^4}{32}-\frac{d^4\mathrm{arccosh}{\left(cx\right)}^2}{4e}-\frac{3d^{2}e\mathrm{arccosh}{\left(cx\right)}^2}{4c^2}-\frac{3e^3\mathrm{arccosh}{\left(cx\right)}^2}{32c^4}+\frac{{(ex+d)}^4\mathrm{arccosh}{\left(cx\right)}^2}{4e}-\frac{2d^3\mathrm{arccosh}\left(cx\right)\sqrt{cx-1}\sqrt{cx+1}}{c}-\frac{4d{e}^2\mathrm{arccosh}\left(cx\right)\sqrt{cx-1}\sqrt{cx+1}}{3c^3}-\frac{3d^{2}ex\mathrm{arccosh}\left(cx\right)\sqrt{cx-1}\sqrt{cx+1}}{2c}-\frac{3e^{3}x\mathrm{arccosh}\left(cx\right)\sqrt{cx-1}\sqrt{cx+1}}{16c^3}-\frac{2d{e}^2{x}^2\mathrm{arccosh}\left(cx\right)\sqrt{cx-1}\sqrt{cx+1}}{3c}-\frac{e^3{x}^3\mathrm{arccosh}\left(cx\right)\sqrt{cx-1}\sqrt{cx+1}}{8c}$$

command

int((e*x+d)^3*arccosh(c*x)^2,x)

Maple 2022.1 output

$$\int {(ex+d)}^3\mathrm{arccosh}{\left(cx\right)}^{2}dx$$

Maple 2021.1 output

$$\frac{72\mathrm{arccosh}{\left(cx\right)}^2{c}^4{x}^4{e}^3+288\mathrm{arccosh}{\left(cx\right)}^2{c}^4{x}^{3}d{e}^2+432\mathrm{arccosh}{\left(cx\right)}^2{c}^4{x}^2{d}^{2}e+288\mathrm{arccosh}{\left(cx\right)}^2{c}^{4}x{d}^3-36\mathrm{arccosh}\left(cx\right)\sqrt{cx-1}\sqrt{cx+1}{c}^3{x}^3{e}^3-192\mathrm{arccosh}\left(cx\right)\sqrt{cx-1}\sqrt{cx+1}{c}^3{x}^{2}d{e}^2-432\mathrm{arccosh}\left(cx\right)\sqrt{cx-1}\sqrt{cx+1}{c}^{3}x{d}^{2}e-576\mathrm{arccosh}\left(cx\right)\sqrt{cx-1}\sqrt{cx+1}{c}^3{d}^3-216\mathrm{arccosh}{\left(cx\right)}^2{c}^2{d}^{2}e-54\mathrm{arccosh}\left(cx\right)\sqrt{cx-1}\sqrt{cx+1}cx{e}^3-384\mathrm{arccosh}\left(cx\right)\sqrt{cx-1}\sqrt{cx+1}cd{e}^2+9c^4{e}^3{x}^4+64c^{4}d{e}^2{x}^3+216c^4{d}^{2}e{x}^2+576x{c}^4{d}^3-27\mathrm{arccosh}{\left(cx\right)}^2{e}^3+27c^2{x}^2{e}^3+384x{c}^{2}d{e}^2}{288c^4}$$