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