\[ \int \frac {\sqrt {1-x^6} \left (1+2 x^6\right )}{1+x^4-2 x^6+x^{12}} \, dx \]
Optimal antiderivative \[ -\frac {\arctan \left (\frac {\sqrt {2}\, x \sqrt {-x^{6}+1}}{x^{6}+x^{2}-1}\right ) \sqrt {2}}{4}-\frac {\arctanh \left (\frac {\sqrt {2}\, x \sqrt {-x^{6}+1}}{x^{6}-x^{2}-1}\right ) \sqrt {2}}{4} \]
command
Integrate[(Sqrt[1 - x^6]*(1 + 2*x^6))/(1 + x^4 - 2*x^6 + x^12),x]
Mathematica 13.1 output
\[ -\frac {\text {ArcTan}\left (\frac {x \sqrt {2-2 x^6}}{-1+x^2+x^6}\right )+\tanh ^{-1}\left (\frac {-1-x^2+x^6}{x \sqrt {2-2 x^6}}\right )}{2 \sqrt {2}} \]
Mathematica 12.3 output
\[ \int \frac {\sqrt {1-x^6} \left (1+2 x^6\right )}{1+x^4-2 x^6+x^{12}} \, dx \]________________________________________________________________________________________