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