\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1+x^3+x^6\right )}{x^6 \left (-1+x^6\right )} \, dx \]
Optimal antiderivative \[ \frac {\left (x^{3}-1\right )^{\frac {2}{3}} \left (3 x^{3}+2\right )}{10 x^{5}}+\frac {\arctan \left (\frac {\sqrt {3}\, x}{x +2^{\frac {2}{3}} \left (x^{3}-1\right )^{\frac {1}{3}}}\right ) 2^{\frac {2}{3}} \sqrt {3}}{6}-\frac {\ln \left (-2 x +2^{\frac {2}{3}} \left (x^{3}-1\right )^{\frac {1}{3}}\right ) 2^{\frac {2}{3}}}{6}+\frac {\ln \left (2 x^{2}+2^{\frac {2}{3}} x \left (x^{3}-1\right )^{\frac {1}{3}}+2^{\frac {1}{3}} \left (x^{3}-1\right )^{\frac {2}{3}}\right ) 2^{\frac {2}{3}}}{12} \]
command
Integrate[((-1 + x^3)^(2/3)*(1 + x^3 + x^6))/(x^6*(-1 + x^6)),x]
Mathematica 13.1 output
\[ \frac {\left (-1+x^3\right )^{2/3} \left (2+3 x^3\right )}{10 x^5}+\frac {\text {ArcTan}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-1+x^3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{-1+x^3}\right )}{3 \sqrt [3]{2}}+\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-1+x^3}+\sqrt [3]{2} \left (-1+x^3\right )^{2/3}\right )}{6 \sqrt [3]{2}} \]
Mathematica 12.3 output
\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1+x^3+x^6\right )}{x^6 \left (-1+x^6\right )} \, dx \]________________________________________________________________________________________