\[ \int \frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{x} \, dx \]
Optimal antiderivative \[ \arctanh \! \left (\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}\right ) \ln \! \left (x \right )-\frac {\arcsinh \! \left (\frac {x \sqrt {e}}{\sqrt {d}}\right )^{2} \sqrt {d}\, \sqrt {1+\frac {e \,x^{2}}{d}}}{2 \sqrt {e \,x^{2}+d}}+\frac {\arcsinh \! \left (\frac {x \sqrt {e}}{\sqrt {d}}\right ) \ln \! \left (1-\left (\frac {x \sqrt {e}}{\sqrt {d}}+\sqrt {1+\frac {e \,x^{2}}{d}}\right )^{2}\right ) \sqrt {d}\, \sqrt {1+\frac {e \,x^{2}}{d}}}{\sqrt {e \,x^{2}+d}}-\frac {\arcsinh \! \left (\frac {x \sqrt {e}}{\sqrt {d}}\right ) \ln \! \left (x \right ) \sqrt {d}\, \sqrt {1+\frac {e \,x^{2}}{d}}}{\sqrt {e \,x^{2}+d}}+\frac {\polylog \! \left (2, \left (\frac {x \sqrt {e}}{\sqrt {d}}+\sqrt {1+\frac {e \,x^{2}}{d}}\right )^{2}\right ) \sqrt {d}\, \sqrt {1+\frac {e \,x^{2}}{d}}}{2 \sqrt {e \,x^{2}+d}} \]
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
\[ \frac {\sqrt {e} \sqrt {\frac {e x^2}{d}+1} \left (-\text {Li}_2\left (e^{-2 \sinh ^{-1}\left (\sqrt {\frac {e}{d}} x\right )}\right )-2 \log (x) \log \left (\sqrt {\frac {e x^2}{d}+1}+x \sqrt {\frac {e}{d}}\right )+\sinh ^{-1}\left (x \sqrt {\frac {e}{d}}\right )^2+2 \sinh ^{-1}\left (x \sqrt {\frac {e}{d}}\right ) \log \left (1-e^{-2 \sinh ^{-1}\left (x \sqrt {\frac {e}{d}}\right )}\right )\right )}{2 \sqrt {\frac {e}{d}} \sqrt {d+e x^2}}+\log (x) \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \]