\[ \int \frac {\left (-1+x^6\right ) \left (1+x^6\right )}{\sqrt [4]{x-x^4+x^7} \left (1+3 x^6+x^{12}\right )} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[((-1 + x^6)*(1 + x^6))/((x - x^4 + x^7)^(1/4)*(1 + 3*x^6 + x^12)),x]
Mathematica 13.1 output
\[ \frac {\sqrt [4]{x} \sqrt [4]{1-x^3+x^6} \text {RootSum}\left [2+2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-3 \log \left (\sqrt [4]{x}\right )+\log \left (\sqrt [4]{1-x^3+x^6}-x^{3/4} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{6 \sqrt [4]{x-x^4+x^7}} \]
Mathematica 12.3 output
\[ \int \frac {\left (-1+x^6\right ) \left (1+x^6\right )}{\sqrt [4]{x-x^4+x^7} \left (1+3 x^6+x^{12}\right )} \, dx \]________________________________________________________________________________________