\[ \int \frac {\left (-1+x^6\right ) \left (1+x^3+x^6\right )^{2/3}}{1+x^6+x^{12}} \, dx \]
Optimal antiderivative \[ -\frac {\arctan \left (\frac {\sqrt {3}\, x}{x +2^{\frac {2}{3}} \left (x^{6}+x^{3}+1\right )^{\frac {1}{3}}}\right ) 2^{\frac {2}{3}} \sqrt {3}}{6}+\frac {\ln \left (-2 x +2^{\frac {2}{3}} \left (x^{6}+x^{3}+1\right )^{\frac {1}{3}}\right ) 2^{\frac {2}{3}}}{6}-\frac {\ln \left (2 x^{2}+2^{\frac {2}{3}} x \left (x^{6}+x^{3}+1\right )^{\frac {1}{3}}+2^{\frac {1}{3}} \left (x^{6}+x^{3}+1\right )^{\frac {2}{3}}\right ) 2^{\frac {2}{3}}}{12} \]
command
Integrate[((-1 + x^6)*(1 + x^3 + x^6)^(2/3))/(1 + x^6 + x^12),x]
Mathematica 13.1 output
\[ -\frac {2 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{1+x^3+x^6}}\right )-2 \log \left (-2 x+2^{2/3} \sqrt [3]{1+x^3+x^6}\right )+\log \left (2 x^2+2^{2/3} x \sqrt [3]{1+x^3+x^6}+\sqrt [3]{2} \left (1+x^3+x^6\right )^{2/3}\right )}{6 \sqrt [3]{2}} \]
Mathematica 12.3 output
\[ \int \frac {\left (-1+x^6\right ) \left (1+x^3+x^6\right )^{2/3}}{1+x^6+x^{12}} \, dx \]________________________________________________________________________________________