\[ \int \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \, dx \]
Optimal antiderivative \[ x \arctanh \left (\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}\right )-\frac {\sqrt {e \,x^{2}+d}}{\sqrt {e}} \]
command
integrate(arctanh(x*e^(1/2)/(e*x^2+d)^(1/2)),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{2} \, x \log \left (-\frac {\frac {\sqrt {e} x}{\sqrt {e x^{2} + d}} + 1}{\frac {\sqrt {e} x}{\sqrt {e x^{2} + d}} - 1}\right ) - \frac {\sqrt {e^{2} x^{2} + d e}}{e} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________