\[ \int \frac {-1+2 x^2+2 x^4}{\left (1+2 x^2\right ) \sqrt {-1+x^6}} \, dx \]
Optimal antiderivative \[ \arctanh \left (\frac {x \left (x^{2}-1\right )}{\sqrt {x^{6}-1}}\right ) \]
command
Integrate[(-1 + 2*x^2 + 2*x^4)/((1 + 2*x^2)*Sqrt[-1 + x^6]),x]
Mathematica 13.1 output
\[ \tanh ^{-1}\left (\frac {x \left (-1+x^2\right )}{\sqrt {-1+x^6}}\right ) \]
Mathematica 12.3 output
\[ \int \frac {-1+2 x^2+2 x^4}{\left (1+2 x^2\right ) \sqrt {-1+x^6}} \, dx \]________________________________________________________________________________________