\[ \int \frac {1-x^2}{\left (1-x+x^2\right ) \left (1-x^3\right )^{2/3}} \, dx \]
Optimal antiderivative \[ -\frac {\ln \left (1+2 \left (1-x \right )^{3}-x^{3}\right ) 2^{\frac {1}{3}}}{4}+\frac {3 \ln \left (2^{\frac {1}{3}} \left (1-x \right )+\left (-x^{3}+1\right )^{\frac {1}{3}}\right ) 2^{\frac {1}{3}}}{4}+\frac {2^{\frac {1}{3}} \arctan \left (\frac {\left (1-\frac {2 \,2^{\frac {1}{3}} \left (1-x \right )}{\left (-x^{3}+1\right )^{\frac {1}{3}}}\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{2} \]
command
Integrate[(1 - x^2)/((1 - x + x^2)*(1 - x^3)^(2/3)),x]
Mathematica 13.1 output
\[ -\frac {2 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \left (1-x^3\right )^{2/3}}{2^{2/3}+2^{2/3} x+2^{2/3} x^2-\left (1-x^3\right )^{2/3}}\right )-2 \log \left (2^{2/3}+2^{2/3} x+2^{2/3} x^2+2 \left (1-x^3\right )^{2/3}\right )+\log \left (-\left (\left (1+x+x^2\right ) \left (\sqrt [3]{2}+\sqrt [3]{2} x^2-\left (2-2 x^3\right )^{2/3}+2 \sqrt [3]{1-x^3}+x \left (\sqrt [3]{2}-2 \sqrt [3]{1-x^3}\right )\right )\right )\right )}{2\ 2^{2/3}} \]
Mathematica 12.3 output
\[ \int \frac {1-x^2}{\left (1-x+x^2\right ) \left (1-x^3\right )^{2/3}} \, dx \]________________________________________________________________________________________