\[ \int \frac {\sqrt [4]{-1+x^4} \left (-1+x^4+2 x^8\right )}{x^6 \left (1-x^4+x^8\right )} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[((-1 + x^4)^(1/4)*(-1 + x^4 + 2*x^8))/(x^6*(1 - x^4 + x^8)),x]
Mathematica 13.1 output
\[ \frac {-4 \left (-1+x^4\right )^{5/4}+15 x^5 \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}}{-1+2 \text {$\#$1}^4}\&\right ]}{20 x^5} \]
Mathematica 12.3 output
\[ \int \frac {\sqrt [4]{-1+x^4} \left (-1+x^4+2 x^8\right )}{x^6 \left (1-x^4+x^8\right )} \, dx \]________________________________________________________________________________________