\[ \int \frac {\left (1+x^6\right ) \sqrt {-2-x^2+x^6}}{4-3 x^4-4 x^6+x^{12}} \, dx \]
Optimal antiderivative \[ -\frac {\sqrt {3+3 \sqrt {3}}\, \arctan \left (\frac {\sqrt {1+\sqrt {3}}\, x}{\sqrt {x^{6}-x^{2}-2}}\right )}{12}-\frac {\sqrt {-3+3 \sqrt {3}}\, \arctanh \left (\frac {\sqrt {\sqrt {3}-1}\, x}{\sqrt {x^{6}-x^{2}-2}}\right )}{12} \]
command
Integrate[((1 + x^6)*Sqrt[-2 - x^2 + x^6])/(4 - 3*x^4 - 4*x^6 + x^12),x]
Mathematica 13.1 output
\[ -\frac {\sqrt {1+\sqrt {3}} \text {ArcTan}\left (\frac {\sqrt {1+\sqrt {3}} x}{\sqrt {-2-x^2+x^6}}\right )+\sqrt {-1+\sqrt {3}} \tanh ^{-1}\left (\frac {\sqrt {-1+\sqrt {3}} x}{\sqrt {-2-x^2+x^6}}\right )}{4 \sqrt {3}} \]
Mathematica 12.3 output
\[ \int \frac {\left (1+x^6\right ) \sqrt {-2-x^2+x^6}}{4-3 x^4-4 x^6+x^{12}} \, dx \]________________________________________________________________________________________