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