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