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