\[ \int \frac {\left (1+x^6\right ) \left (-1+x^3+x^6\right )^{2/3}}{1-x^6+x^{12}} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[((1 + x^6)*(-1 + x^3 + x^6)^(2/3))/(1 - x^6 + x^12),x]
Mathematica 13.1 output
\[ \frac {1}{6} \text {RootSum}\left [2-2 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}^2+\log \left (\sqrt [3]{-1+x^3+x^6}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-1+\text {$\#$1}^3}\&\right ] \]
Mathematica 12.3 output
\[ \int \frac {\left (1+x^6\right ) \left (-1+x^3+x^6\right )^{2/3}}{1-x^6+x^{12}} \, dx \]________________________________________________________________________________________