\[ \int \frac {\left (-2+x^6\right ) \left (4+x^6\right ) \sqrt [4]{-2+2 x^4+x^6}}{x^6 \left (-4-x^4+2 x^6\right )} \, dx \]
Optimal antiderivative \[ \frac {\left (x^{6}+2 x^{4}-2\right )^{\frac {1}{4}} \left (2 x^{6}+9 x^{4}-4\right )}{10 x^{5}}+\frac {5^{\frac {1}{4}} 2^{\frac {3}{4}} \arctan \left (\frac {5^{\frac {1}{4}} 2^{\frac {3}{4}} x}{2 \left (x^{6}+2 x^{4}-2\right )^{\frac {1}{4}}}\right )}{8}-\frac {5^{\frac {1}{4}} 2^{\frac {3}{4}} \arctanh \left (\frac {5^{\frac {1}{4}} 2^{\frac {3}{4}} x}{2 \left (x^{6}+2 x^{4}-2\right )^{\frac {1}{4}}}\right )}{8} \]
command
Integrate[((-2 + x^6)*(4 + x^6)*(-2 + 2*x^4 + x^6)^(1/4))/(x^6*(-4 - x^4 + 2*x^6)),x]
Mathematica 13.1 output
\[ \frac {\sqrt [4]{-2+2 x^4+x^6} \left (-4+9 x^4+2 x^6\right )}{10 x^5}+\frac {1}{4} \sqrt [4]{\frac {5}{2}} \text {ArcTan}\left (\frac {\sqrt [4]{\frac {5}{2}} x}{\sqrt [4]{-2+2 x^4+x^6}}\right )-\frac {1}{4} \sqrt [4]{\frac {5}{2}} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {5}{2}} x}{\sqrt [4]{-2+2 x^4+x^6}}\right ) \]
Mathematica 12.3 output
\[ \int \frac {\left (-2+x^6\right ) \left (4+x^6\right ) \sqrt [4]{-2+2 x^4+x^6}}{x^6 \left (-4-x^4+2 x^6\right )} \, dx \]________________________________________________________________________________________