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