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