\[ \int \frac {\sqrt {-1-x^2+x^6} \left (1+2 x^6\right )}{8-x^4-16 x^6+8 x^{12}} \, dx \]
Optimal antiderivative \[ \frac {\sqrt {8-2 \sqrt {2}}\, \arctan \left (\frac {\sqrt {4-\sqrt {2}}\, x}{2 \sqrt {x^{6}-x^{2}-1}}\right )}{16}-\frac {\sqrt {8+2 \sqrt {2}}\, \arctan \left (\frac {\sqrt {4+\sqrt {2}}\, x}{2 \sqrt {x^{6}-x^{2}-1}}\right )}{16} \]
command
Integrate[(Sqrt[-1 - x^2 + x^6]*(1 + 2*x^6))/(8 - x^4 - 16*x^6 + 8*x^12),x]
Mathematica 13.1 output
\[ \frac {1}{16} \left (\sqrt {8-2 \sqrt {2}} \text {ArcTan}\left (\frac {x}{2 \sqrt {\frac {1+x^2-x^6}{-4+\sqrt {2}}}}\right )-\sqrt {2 \left (4+\sqrt {2}\right )} \text {ArcTan}\left (\frac {\sqrt {4+\sqrt {2}} x}{2 \sqrt {-1-x^2+x^6}}\right )\right ) \]
Mathematica 12.3 output
\[ \int \frac {\sqrt {-1-x^2+x^6} \left (1+2 x^6\right )}{8-x^4-16 x^6+8 x^{12}} \, dx \]________________________________________________________________________________________