\[ \int \frac {1+x}{\left (1+3 x+x^2\right ) \sqrt [3]{1-x^3}} \, dx \]
Optimal antiderivative \[ \frac {\sqrt {3}\, \arctan \left (\frac {5 \sqrt {3}\, \left (-x^{3}+1\right )^{\frac {1}{3}}}{2 \,2^{\frac {1}{3}} 5^{\frac {2}{3}}-2 \,2^{\frac {1}{3}} 5^{\frac {2}{3}} x +5 \left (-x^{3}+1\right )^{\frac {1}{3}}}\right ) 2^{\frac {2}{3}} 5^{\frac {1}{3}}}{10}+\frac {\ln \left (-2^{\frac {1}{3}} 5^{\frac {2}{3}}+2^{\frac {1}{3}} 5^{\frac {2}{3}} x +5 \left (-x^{3}+1\right )^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} 5^{\frac {1}{3}}}{10}-\frac {\ln \left (2^{\frac {2}{3}} 5^{\frac {1}{3}}-2 \,2^{\frac {2}{3}} 5^{\frac {1}{3}} x +2^{\frac {2}{3}} 5^{\frac {1}{3}} x^{2}+\left (2^{\frac {1}{3}} 5^{\frac {2}{3}}-2^{\frac {1}{3}} 5^{\frac {2}{3}} x \right ) \left (-x^{3}+1\right )^{\frac {1}{3}}+5 \left (-x^{3}+1\right )^{\frac {2}{3}}\right ) 2^{\frac {2}{3}} 5^{\frac {1}{3}}}{20} \]
command
Integrate[(1 + x)/((1 + 3*x + x^2)*(1 - x^3)^(1/3)),x]
Mathematica 13.1 output
\[ \frac {2 \sqrt {3} \text {ArcTan}\left (\frac {5 \sqrt {3} \sqrt [3]{1-x^3}}{2 \sqrt [3]{2} 5^{2/3}-2 \sqrt [3]{2} 5^{2/3} x+5 \sqrt [3]{1-x^3}}\right )+2 \log \left (-\sqrt [3]{2} 5^{2/3}+\sqrt [3]{2} 5^{2/3} x+5 \sqrt [3]{1-x^3}\right )-\log \left (2^{2/3} \sqrt [3]{5}-2\ 2^{2/3} \sqrt [3]{5} x+2^{2/3} \sqrt [3]{5} x^2-5^{2/3} (-1+x) \sqrt [3]{2-2 x^3}+5 \left (1-x^3\right )^{2/3}\right )}{2 \sqrt [3]{2} 5^{2/3}} \]
Mathematica 12.3 output
\[ \int \frac {1+x}{\left (1+3 x+x^2\right ) \sqrt [3]{1-x^3}} \, dx \]________________________________________________________________________________________