\[ \int \frac {\sqrt {-1-x^2+x^6} \left (1+2 x^6\right )}{\left (-1+x^6\right ) \left (-2+x^2+2 x^6\right )} \, dx \]
Optimal antiderivative \[ \arctan \left (\frac {x}{\sqrt {x^{6}-x^{2}-1}}\right )-\frac {\sqrt {6}\, \arctan \left (\frac {\sqrt {6}\, x}{2 \sqrt {x^{6}-x^{2}-1}}\right )}{2} \]
command
Integrate[(Sqrt[-1 - x^2 + x^6]*(1 + 2*x^6))/((-1 + x^6)*(-2 + x^2 + 2*x^6)),x]
Mathematica 13.1 output
\[ \text {ArcTan}\left (\frac {x}{\sqrt {-1-x^2+x^6}}\right )-\sqrt {\frac {3}{2}} \text {ArcTan}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt {-1-x^2+x^6}}\right ) \]
Mathematica 12.3 output
\[ \int \frac {\sqrt {-1-x^2+x^6} \left (1+2 x^6\right )}{\left (-1+x^6\right ) \left (-2+x^2+2 x^6\right )} \, dx \]________________________________________________________________________________________