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