6.1 Problem number 109

$$\int \frac{a+b{\csc }^{-1}(cx)}{{(d+e{x}^2)}^2}dx$$

Optimal antiderivative $$\frac{(a+b\mathrm{arccsc}\left(cx\right))\ln (1-\frac{\mathrm{I}c(\frac{\mathrm{I}}{cx}+\sqrt{1-\frac{1}{c^2{x}^2}})\sqrt{-d}}{\sqrt{e}-\sqrt{c^{2}d+e}})}{4{(-d)}^{\frac{3}{2}}\sqrt{e}}-\frac{(a+b\mathrm{arccsc}\left(cx\right))\ln (1+\frac{\mathrm{I}c(\frac{\mathrm{I}}{cx}+\sqrt{1-\frac{1}{c^2{x}^2}})\sqrt{-d}}{\sqrt{e}-\sqrt{c^{2}d+e}})}{4{(-d)}^{\frac{3}{2}}\sqrt{e}}+\frac{(a+b\mathrm{arccsc}\left(cx\right))\ln (1-\frac{\mathrm{I}c(\frac{\mathrm{I}}{cx}+\sqrt{1-\frac{1}{c^2{x}^2}})\sqrt{-d}}{\sqrt{e}+\sqrt{c^{2}d+e}})}{4{(-d)}^{\frac{3}{2}}\sqrt{e}}-\frac{(a+b\mathrm{arccsc}\left(cx\right))\ln (1+\frac{\mathrm{I}c(\frac{\mathrm{I}}{cx}+\sqrt{1-\frac{1}{c^2{x}^2}})\sqrt{-d}}{\sqrt{e}+\sqrt{c^{2}d+e}})}{4{(-d)}^{\frac{3}{2}}\sqrt{e}}+\frac{\mathrm{I}b\mathrm{polylog}(2,\frac{-\mathrm{I}c(\frac{\mathrm{I}}{cx}+\sqrt{1-\frac{1}{c^2{x}^2}})\sqrt{-d}}{\sqrt{e}-\sqrt{c^{2}d+e}})}{4{(-d)}^{\frac{3}{2}}\sqrt{e}}-\frac{\mathrm{I}b\mathrm{polylog}(2,\frac{\mathrm{I}c(\frac{\mathrm{I}}{cx}+\sqrt{1-\frac{1}{c^2{x}^2}})\sqrt{-d}}{\sqrt{e}-\sqrt{c^{2}d+e}})}{4{(-d)}^{\frac{3}{2}}\sqrt{e}}+\frac{\mathrm{I}b\mathrm{polylog}(2,\frac{-\mathrm{I}c(\frac{\mathrm{I}}{cx}+\sqrt{1-\frac{1}{c^2{x}^2}})\sqrt{-d}}{\sqrt{e}+\sqrt{c^{2}d+e}})}{4{(-d)}^{\frac{3}{2}}\sqrt{e}}-\frac{\mathrm{I}b\mathrm{polylog}(2,\frac{\mathrm{I}c(\frac{\mathrm{I}}{cx}+\sqrt{1-\frac{1}{c^2{x}^2}})\sqrt{-d}}{\sqrt{e}+\sqrt{c^{2}d+e}})}{4{(-d)}^{\frac{3}{2}}\sqrt{e}}+\frac{-a-b\mathrm{arccsc}\left(cx\right)}{4d(-\frac{d}{x}+\sqrt{-d}\sqrt{e})}+\frac{a+b\mathrm{arccsc}\left(cx\right)}{4d(\frac{d}{x}+\sqrt{-d}\sqrt{e})}+\frac{b\mathrm{arctanh}\left(\frac{c^{2}d-\frac{\sqrt{-d}\sqrt{e}}{x}}{c\sqrt{d}\sqrt{c^{2}d+e}\sqrt{1-\frac{1}{c^2{x}^2}}}\right)}{4d^{\frac{3}{2}}\sqrt{c^{2}d+e}}+\frac{b\mathrm{arctanh}\left(\frac{c^{2}d+\frac{\sqrt{-d}\sqrt{e}}{x}}{c\sqrt{d}\sqrt{c^{2}d+e}\sqrt{1-\frac{1}{c^2{x}^2}}}\right)}{4d^{\frac{3}{2}}\sqrt{c^{2}d+e}}$$

command

int((a+b*arccsc(c*x))/(e*x^2+d)^2,x)

Maple 2022.1 output

hanged

Maple 2021.1 output

$$\text{output too large to display}$$