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