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