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