\[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (-2-x^3+x^6\right )} \, dx \]
Optimal antiderivative \[ \frac {\left (x^{3}+1\right )^{\frac {5}{3}}}{5 x^{5}}-\frac {\arctan \left (\frac {3^{\frac {5}{6}} x}{3^{\frac {1}{3}} x +2 \,2^{\frac {1}{3}} \left (x^{3}+1\right )^{\frac {1}{3}}}\right ) 2^{\frac {1}{3}} 3^{\frac {1}{6}}}{6}+\frac {\ln \left (-3 x +2^{\frac {1}{3}} 3^{\frac {2}{3}} \left (x^{3}+1\right )^{\frac {1}{3}}\right ) 2^{\frac {1}{3}} 3^{\frac {2}{3}}}{18}-\frac {\ln \left (3 x^{2}+2^{\frac {1}{3}} 3^{\frac {2}{3}} x \left (x^{3}+1\right )^{\frac {1}{3}}+2^{\frac {2}{3}} 3^{\frac {1}{3}} \left (x^{3}+1\right )^{\frac {2}{3}}\right ) 2^{\frac {1}{3}} 3^{\frac {2}{3}}}{36} \]
command
Integrate[((1 + x^3)^(2/3)*(2 + x^3))/(x^6*(-2 - x^3 + x^6)),x]
Mathematica 13.1 output
\[ \frac {\left (1+x^3\right )^{5/3}}{5 x^5}-\frac {\text {ArcTan}\left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{2} \sqrt [3]{1+x^3}}\right )}{2^{2/3} 3^{5/6}}+\frac {\log \left (-3 x+\sqrt [3]{2} 3^{2/3} \sqrt [3]{1+x^3}\right )}{3\ 2^{2/3} \sqrt [3]{3}}-\frac {\log \left (3 x^2+\sqrt [3]{2} 3^{2/3} x \sqrt [3]{1+x^3}+2^{2/3} \sqrt [3]{3} \left (1+x^3\right )^{2/3}\right )}{6\ 2^{2/3} \sqrt [3]{3}} \]
Mathematica 12.3 output
\[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (-2-x^3+x^6\right )} \, dx \]________________________________________________________________________________________