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