\[ \int \frac {1}{(1+x) \sqrt [3]{1-x^3}} \, dx \]
Optimal antiderivative \[ -\frac {\ln \left (\left (1-x \right ) \left (1+x \right )^{2}\right ) 2^{\frac {2}{3}}}{8}+\frac {3 \ln \left (-1+x +2^{\frac {2}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}}\right ) 2^{\frac {2}{3}}}{8}-\frac {\arctan \left (\frac {\left (1+\frac {2^{\frac {1}{3}} \left (1-x \right )}{\left (-x^{3}+1\right )^{\frac {1}{3}}}\right ) \sqrt {3}}{3}\right ) \sqrt {3}\, 2^{\frac {2}{3}}}{4} \]
command
Integrate[1/((1 + x)*(1 - x^3)^(1/3)),x]
Mathematica 13.1 output
\[ \frac {2 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{1-x^3}}{\sqrt [3]{2}-\sqrt [3]{2} x+\sqrt [3]{1-x^3}}\right )+2 \log \left (-\sqrt [3]{2}+\sqrt [3]{2} x+2 \sqrt [3]{1-x^3}\right )-\log \left (2^{2/3}-2\ 2^{2/3} x+2^{2/3} x^2-2 (-1+x) \sqrt [3]{2-2 x^3}+4 \left (1-x^3\right )^{2/3}\right )}{4 \sqrt [3]{2}} \]
Mathematica 12.3 output
\[ \int \frac {1}{(1+x) \sqrt [3]{1-x^3}} \, dx \]________________________________________________________________________________________