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