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