\[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right ) \left (-2+x^3+2 x^6\right )}{x^6 \left (-1-x^3+x^6\right )} \, dx \]
Optimal antiderivative \[ \frac {\left (x^{6}-1\right )^{\frac {2}{3}} \left (4 x^{6}+15 x^{3}-4\right )}{10 x^{5}}-\arctan \left (\frac {\sqrt {3}\, x}{x +2 \left (x^{6}-1\right )^{\frac {1}{3}}}\right ) \sqrt {3}+\ln \left (-x +\left (x^{6}-1\right )^{\frac {1}{3}}\right )-\frac {\ln \left (x^{2}+x \left (x^{6}-1\right )^{\frac {1}{3}}+\left (x^{6}-1\right )^{\frac {2}{3}}\right )}{2} \]
command
Integrate[((-1 + x^6)^(2/3)*(1 + x^6)*(-2 + x^3 + 2*x^6))/(x^6*(-1 - x^3 + x^6)),x]
Mathematica 13.1 output
\[ \frac {\left (-1+x^6\right )^{2/3} \left (-4+15 x^3+4 x^6\right )}{10 x^5}-\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^6}}\right )+\log \left (-x+\sqrt [3]{-1+x^6}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \]
Mathematica 12.3 output
\[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right ) \left (-2+x^3+2 x^6\right )}{x^6 \left (-1-x^3+x^6\right )} \, dx \]________________________________________________________________________________________