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