\[ \int \frac {1+x}{\left (1+4 x+x^2\right ) \sqrt [3]{1-x^3}} \, dx \]
Optimal antiderivative \[ \frac {\arctan \left (\frac {\sqrt {3}\, \left (-x^{3}+1\right )^{\frac {1}{3}}}{2^{\frac {2}{3}}-2^{\frac {2}{3}} x +\left (-x^{3}+1\right )^{\frac {1}{3}}}\right ) 2^{\frac {1}{3}} \sqrt {3}}{6}+\frac {\ln \left (-2^{\frac {2}{3}}+2^{\frac {2}{3}} x +2 \left (-x^{3}+1\right )^{\frac {1}{3}}\right ) 2^{\frac {1}{3}}}{6}-\frac {\ln \left (-2^{\frac {1}{3}}+2 \,2^{\frac {1}{3}} x -2^{\frac {1}{3}} x^{2}+\left (-2^{\frac {2}{3}}+2^{\frac {2}{3}} x \right ) \left (-x^{3}+1\right )^{\frac {1}{3}}-2 \left (-x^{3}+1\right )^{\frac {2}{3}}\right ) 2^{\frac {1}{3}}}{12} \]
command
Integrate[(1 + x)/((1 + 4*x + x^2)*(1 - x^3)^(1/3)),x]
Mathematica 13.1 output
\[ \frac {2 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{1-x^3}}{2^{2/3}-2^{2/3} x+\sqrt [3]{1-x^3}}\right )+2 \log \left (-2^{2/3}+2^{2/3} x+2 \sqrt [3]{1-x^3}\right )-\log \left (-\sqrt [3]{2}+2 \sqrt [3]{2} x-\sqrt [3]{2} x^2+2^{2/3} (-1+x) \sqrt [3]{1-x^3}-2 \left (1-x^3\right )^{2/3}\right )}{6\ 2^{2/3}} \]
Mathematica 12.3 output
\[ \int \frac {1+x}{\left (1+4 x+x^2\right ) \sqrt [3]{1-x^3}} \, dx \]________________________________________________________________________________________