\[ \int \frac {(-3+2 x) \sqrt [3]{-1+x+x^3}}{x^2 \left (2-2 x+x^3\right )} \, dx \]
Optimal antiderivative \[ \frac {3 \left (x^{3}+x -1\right )^{\frac {1}{3}}}{2 x}+\frac {3^{\frac {5}{6}} \arctan \left (\frac {3^{\frac {5}{6}} x}{3^{\frac {1}{3}} x +2 \,2^{\frac {1}{3}} \left (x^{3}+x -1\right )^{\frac {1}{3}}}\right ) 2^{\frac {2}{3}}}{4}+\frac {2^{\frac {2}{3}} 3^{\frac {1}{3}} \ln \left (-3 x +2^{\frac {1}{3}} 3^{\frac {2}{3}} \left (x^{3}+x -1\right )^{\frac {1}{3}}\right )}{4}-\frac {2^{\frac {2}{3}} 3^{\frac {1}{3}} \ln \left (3 x^{2}+2^{\frac {1}{3}} 3^{\frac {2}{3}} x \left (x^{3}+x -1\right )^{\frac {1}{3}}+2^{\frac {2}{3}} 3^{\frac {1}{3}} \left (x^{3}+x -1\right )^{\frac {2}{3}}\right )}{8} \]
command
Integrate[((-3 + 2*x)*(-1 + x + x^3)^(1/3))/(x^2*(2 - 2*x + x^3)),x]
Mathematica 13.1 output
\[ \frac {3 \sqrt [3]{-1+x+x^3}}{2 x}+\frac {3^{5/6} \text {ArcTan}\left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{2} \sqrt [3]{-1+x+x^3}}\right )}{2 \sqrt [3]{2}}+\frac {1}{2} \sqrt [3]{\frac {3}{2}} \log \left (-3 x+\sqrt [3]{2} 3^{2/3} \sqrt [3]{-1+x+x^3}\right )-\frac {1}{4} \sqrt [3]{\frac {3}{2}} \log \left (3 x^2+\sqrt [3]{2} 3^{2/3} x \sqrt [3]{-1+x+x^3}+2^{2/3} \sqrt [3]{3} \left (-1+x+x^3\right )^{2/3}\right ) \]
Mathematica 12.3 output
\[ \int \frac {(-3+2 x) \sqrt [3]{-1+x+x^3}}{x^2 \left (2-2 x+x^3\right )} \, dx \]________________________________________________________________________________________