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