\[ \int \frac {\left (2+x^6\right ) \left (-1-x^4+x^6\right )}{\sqrt [4]{1-x^4-x^6} \left (-1+x^6\right )^2} \, dx \]
Optimal antiderivative \[ -\frac {x \left (-x^{6}-x^{4}+1\right )^{\frac {3}{4}}}{2 x^{6}-2}-\frac {5 \arctan \left (\frac {\sqrt {2}\, x \left (-x^{6}-x^{4}+1\right )^{\frac {1}{4}}}{-x^{2}+\sqrt {-x^{6}-x^{4}+1}}\right ) \sqrt {2}}{8}-\frac {5 \arctanh \left (\frac {\sqrt {2}\, x \left (-x^{6}-x^{4}+1\right )^{\frac {1}{4}}}{x^{2}+\sqrt {-x^{6}-x^{4}+1}}\right ) \sqrt {2}}{8} \]
command
Integrate[((2 + x^6)*(-1 - x^4 + x^6))/((1 - x^4 - x^6)^(1/4)*(-1 + x^6)^2),x]
Mathematica 13.1 output
\[ -\frac {x \left (1-x^4-x^6\right )^{3/4}}{2 \left (-1+x^6\right )}-\frac {5 \text {ArcTan}\left (\frac {\sqrt {2} x \sqrt [4]{1-x^4-x^6}}{-x^2+\sqrt {1-x^4-x^6}}\right )}{4 \sqrt {2}}-\frac {5 \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{1-x^4-x^6}}{x^2+\sqrt {1-x^4-x^6}}\right )}{4 \sqrt {2}} \]
Mathematica 12.3 output
\[ \int \frac {\left (2+x^6\right ) \left (-1-x^4+x^6\right )}{\sqrt [4]{1-x^4-x^6} \left (-1+x^6\right )^2} \, dx \]________________________________________________________________________________________