\[ \int \frac {\left (1+x^6\right )^2 \left (-1+2 x^6\right )}{\left (1-x^2+x^6\right )^{3/2} \left (1-x^2-x^4+2 x^6-x^8+x^{12}\right )} \, dx \]
Optimal antiderivative \[ \frac {x}{\sqrt {x^{6}-x^{2}+1}}-\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2+2 \sqrt {5}}\, x}{2 \sqrt {x^{6}-x^{2}+1}}\right )}{\sqrt {145+65 \sqrt {5}}}-\frac {\sqrt {290+130 \sqrt {5}}\, \arctanh \left (\frac {\sqrt {-2+2 \sqrt {5}}\, x}{2 \sqrt {x^{6}-x^{2}+1}}\right )}{10} \]
command
Integrate[((1 + x^6)^2*(-1 + 2*x^6))/((1 - x^2 + x^6)^(3/2)*(1 - x^2 - x^4 + 2*x^6 - x^8 + x^12)),x]
Mathematica 13.1 output
\[ \frac {x}{\sqrt {1-x^2+x^6}}-\sqrt {\frac {2}{145+65 \sqrt {5}}} \text {ArcTan}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} x}{\sqrt {1-x^2+x^6}}\right )-\sqrt {\frac {1}{10} \left (29+13 \sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} x}{\sqrt {1-x^2+x^6}}\right ) \]
Mathematica 12.3 output
\[ \int \frac {\left (1+x^6\right )^2 \left (-1+2 x^6\right )}{\left (1-x^2+x^6\right )^{3/2} \left (1-x^2-x^4+2 x^6-x^8+x^{12}\right )} \, dx \]________________________________________________________________________________________