19.1 Problem number 4

$$\int \frac{{\tanh }^{-1}\left(\frac{\sqrt{e}x}{\sqrt{d+e{x}^2}}\right)}{x}dx$$

Optimal antiderivative $$\mathrm{arctanh}\left(\frac{x\sqrt{e}}{\sqrt{e{x}^2+d}}\right)\ln \left(x\right)-\frac{\mathrm{arcsinh}{\left(\frac{x\sqrt{e}}{\sqrt{d}}\right)}^2\sqrt{d}\sqrt{1+\frac{e{x}^2}{d}}}{2\sqrt{e{x}^2+d}}+\frac{\mathrm{arcsinh}\left(\frac{x\sqrt{e}}{\sqrt{d}}\right)\ln (1-{(\frac{x\sqrt{e}}{\sqrt{d}}+\sqrt{1+\frac{e{x}^2}{d}})}^2)\sqrt{d}\sqrt{1+\frac{e{x}^2}{d}}}{\sqrt{e{x}^2+d}}-\frac{\mathrm{arcsinh}\left(\frac{x\sqrt{e}}{\sqrt{d}}\right)\ln \left(x\right)\sqrt{d}\sqrt{1+\frac{e{x}^2}{d}}}{\sqrt{e{x}^2+d}}+\frac{\mathrm{polylog}(2,{(\frac{x\sqrt{e}}{\sqrt{d}}+\sqrt{1+\frac{e{x}^2}{d}})}^2)\sqrt{d}\sqrt{1+\frac{e{x}^2}{d}}}{2\sqrt{e{x}^2+d}}$$

command

Integrate[ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]]/x,x]

Mathematica 13.1 output

$$\int \frac{{\tanh }^{-1}\left(\frac{\sqrt{e}x}{\sqrt{d+e{x}^2}}\right)}{x}dx$$

Mathematica 12.3 output

$$\frac{\sqrt{e}\sqrt{\frac{e{x}^2}{d}+1}(-{\text{Li}}_2\left(e^{-2{\sinh }^{-1}\left(\sqrt{\frac{e}{d}}x\right)}\right)-2\log (x)\log (\sqrt{\frac{e{x}^2}{d}+1}+x\sqrt{\frac{e}{d}})+{\sinh }^{-1}{\left(x\sqrt{\frac{e}{d}}\right)}^2+2{\sinh }^{-1}\left(x\sqrt{\frac{e}{d}}\right)\log (1-e^{-2{\sinh }^{-1}\left(x\sqrt{\frac{e}{d}}\right)}))}{2\sqrt{\frac{e}{d}}\sqrt{d+e{x}^2}}+\log (x){\tanh }^{-1}\left(\frac{\sqrt{e}x}{\sqrt{d+e{x}^2}}\right)$$