\[ \int \frac {\left (-1+x^6\right ) \left (1+x^6\right )^{2/3}}{x^3 \left (2-x^3+2 x^6\right )} \, dx \]
Optimal antiderivative \[ \frac {\left (x^{6}+1\right )^{\frac {2}{3}}}{4 x^{2}}-\frac {\arctan \left (\frac {\sqrt {3}\, x}{x +2 \,2^{\frac {1}{3}} \left (x^{6}+1\right )^{\frac {1}{3}}}\right ) 2^{\frac {1}{3}} \sqrt {3}}{12}+\frac {\ln \left (-x +2^{\frac {1}{3}} \left (x^{6}+1\right )^{\frac {1}{3}}\right ) 2^{\frac {1}{3}}}{12}-\frac {\ln \left (x^{2}+2^{\frac {1}{3}} x \left (x^{6}+1\right )^{\frac {1}{3}}+2^{\frac {2}{3}} \left (x^{6}+1\right )^{\frac {2}{3}}\right ) 2^{\frac {1}{3}}}{24} \]
command
Integrate[((-1 + x^6)*(1 + x^6)^(2/3))/(x^3*(2 - x^3 + 2*x^6)),x]
Mathematica 13.1 output
\[ \frac {1}{24} \left (\frac {6 \left (1+x^6\right )^{2/3}}{x^2}-2 \sqrt [3]{2} \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{1+x^6}}\right )+2 \sqrt [3]{2} \log \left (-x+\sqrt [3]{2} \sqrt [3]{1+x^6}\right )-\sqrt [3]{2} \log \left (x^2+\sqrt [3]{2} x \sqrt [3]{1+x^6}+2^{2/3} \left (1+x^6\right )^{2/3}\right )\right ) \]
Mathematica 12.3 output
\[ \int \frac {\left (-1+x^6\right ) \left (1+x^6\right )^{2/3}}{x^3 \left (2-x^3+2 x^6\right )} \, dx \]________________________________________________________________________________________