13.1 Problem number 62

$$\int \frac{\text{ArcTan}(d+ex)}{a+bx+c{x}^2}dx$$

Optimal antiderivative $$\frac{\arctan (ex+d)\ln \left(\frac{2e(b+2cx-\sqrt{-4ac+b^2})}{(1-\mathrm{I}(ex+d))(2c(\mathrm{I}-d)+e(b-\sqrt{-4ac+b^2}))}\right)}{\sqrt{-4ac+b^2}}-\frac{\arctan (ex+d)\ln \left(\frac{2e(b+2cx+\sqrt{-4ac+b^2})}{(1-\mathrm{I}(ex+d))(2c(\mathrm{I}-d)+e(b+\sqrt{-4ac+b^2}))}\right)}{\sqrt{-4ac+b^2}}-\frac{\mathrm{I}\mathrm{polylog}(2,1+\frac{4cd-4c(ex+d)-2e(b-\sqrt{-4ac+b^2})}{(1-\mathrm{I}(ex+d))(2\mathrm{I}c-2cd+be-e\sqrt{-4ac+b^2})})}{2\sqrt{-4ac+b^2}}+\frac{\mathrm{I}\mathrm{polylog}(2,1+\frac{4cd-4c(ex+d)-2e(b+\sqrt{-4ac+b^2})}{(1-\mathrm{I}(ex+d))(2c(\mathrm{I}-d)+e(b+\sqrt{-4ac+b^2}))})}{2\sqrt{-4ac+b^2}}$$

command

Integrate[ArcTan[d + e*x]/(a + b*x + c*x^2),x]

Mathematica 13.1 output

$$\text{\$Aborted}$$

Mathematica 12.3 output

$$\frac{i(-{\text{Li}}_2\left(\frac{2c(d+ex-i)}{2c(d-i)+(\sqrt{b^2-4ac}-b)e}\right)+{\text{Li}}_2\left(\frac{2c(d+ex-i)}{2c(d-i)-(b+\sqrt{b^2-4ac})e}\right)+{\text{Li}}_2\left(\frac{2c(d+ex+i)}{2c(d+i)+(\sqrt{b^2-4ac}-b)e}\right)-{\text{Li}}_2\left(\frac{2c(d+ex+i)}{2c(d+i)-(b+\sqrt{b^2-4ac})e}\right)+\log (1-i(d+ex))\log \left(\frac{e(\sqrt{b^2-4ac}-b-2cx)}{e(\sqrt{b^2-4ac}-b)+2c(d+i)}\right)-\log (1-i(d+ex))\log \left(\frac{e(\sqrt{b^2-4ac}+b+2cx)}{e(\sqrt{b^2-4ac}+b)-2c(d+i)}\right)-\log (1+i(d+ex))\log \left(\frac{e(\sqrt{b^2-4ac}-b-2cx)}{e(\sqrt{b^2-4ac}-b)+2c(d-i)}\right)+\log (1+i(d+ex))\log \left(\frac{e(\sqrt{b^2-4ac}+b+2cx)}{e(\sqrt{b^2-4ac}+b)-2c(d-i)}\right))}{2\sqrt{b^2-4ac}}$$