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