\[ \int \frac {\left (2+x^6\right ) \left (1-2 x^6+x^8+x^{12}\right )}{x^8 \sqrt [4]{-1+x^6} \left (-1+x^4+x^6\right )} \, dx \]
Optimal antiderivative \[ \frac {2 \left (x^{6}-1\right )^{\frac {3}{4}} \left (3 x^{6}-7 x^{4}-3\right )}{21 x^{7}}-\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x \left (x^{6}-1\right )^{\frac {1}{4}}}{-x^{2}+\sqrt {x^{6}-1}}\right )-\sqrt {2}\, \arctanh \left (\frac {\sqrt {2}\, x \left (x^{6}-1\right )^{\frac {1}{4}}}{x^{2}+\sqrt {x^{6}-1}}\right ) \]
command
Integrate[((2 + x^6)*(1 - 2*x^6 + x^8 + x^12))/(x^8*(-1 + x^6)^(1/4)*(-1 + x^4 + x^6)),x]
Mathematica 13.1 output
\[ \frac {2 \left (-1+x^6\right )^{3/4} \left (-3-7 x^4+3 x^6\right )}{21 x^7}-\sqrt {2} \text {ArcTan}\left (\frac {\sqrt {2} x \sqrt [4]{-1+x^6}}{-x^2+\sqrt {-1+x^6}}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{-1+x^6}}{x^2+\sqrt {-1+x^6}}\right ) \]
Mathematica 12.3 output
\[ \int \frac {\left (2+x^6\right ) \left (1-2 x^6+x^8+x^{12}\right )}{x^8 \sqrt [4]{-1+x^6} \left (-1+x^4+x^6\right )} \, dx \]________________________________________________________________________________________