\[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^2 \left (-1+x^3+x^6\right )} \, dx \]
Optimal antiderivative \[ \frac {\left (x^{6}-1\right )^{\frac {1}{3}}}{x}+\frac {\arctan \! \left (\frac {\sqrt {3}\, x}{-x +2 \left (x^{6}-1\right )^{\frac {1}{3}}}\right ) \sqrt {3}}{3}-\frac {\ln \! \left (x +\left (x^{6}-1\right )^{\frac {1}{3}}\right )}{3}+\frac {\ln \! \left (x^{2}-x \left (x^{6}-1\right )^{\frac {1}{3}}+\left (x^{6}-1\right )^{\frac {2}{3}}\right )}{6} \]
command
Int[((-1 + x^6)^(1/3)*(1 + x^6))/(x^2*(-1 + x^3 + x^6)),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^2 \left (-1+x^3+x^6\right )} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ -\frac {2 \left (5-\sqrt {5}\right ) \sqrt [3]{x^6-1} x^5 \operatorname {AppellF1}\left (\frac {5}{6},-\frac {1}{3},1,\frac {11}{6},x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{25 \left (3-\sqrt {5}\right ) \sqrt [3]{1-x^6}}-\frac {2 \sqrt [3]{x^6-1} x^5 \operatorname {AppellF1}\left (\frac {5}{6},-\frac {1}{3},1,\frac {11}{6},x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{5 \sqrt {5} \left (3-\sqrt {5}\right ) \sqrt [3]{1-x^6}}-\frac {2 \left (5+\sqrt {5}\right ) \sqrt [3]{x^6-1} x^5 \operatorname {AppellF1}\left (\frac {5}{6},-\frac {1}{3},1,\frac {11}{6},x^6,\frac {2 x^6}{3+\sqrt {5}}\right )}{25 \left (3+\sqrt {5}\right ) \sqrt [3]{1-x^6}}+\frac {2 \sqrt [3]{x^6-1} x^5 \operatorname {AppellF1}\left (\frac {5}{6},-\frac {1}{3},1,\frac {11}{6},x^6,\frac {2 x^6}{3+\sqrt {5}}\right )}{5 \sqrt {5} \left (3+\sqrt {5}\right ) \sqrt [3]{1-x^6}}-\frac {\left (5-\sqrt {5}\right ) \sqrt [3]{x^6-1} x^2 \operatorname {AppellF1}\left (\frac {1}{3},-\frac {1}{3},1,\frac {4}{3},x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{10 \left (3-\sqrt {5}\right ) \sqrt [3]{1-x^6}}-\frac {\sqrt [3]{x^6-1} x^2 \operatorname {AppellF1}\left (\frac {1}{3},-\frac {1}{3},1,\frac {4}{3},x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{\sqrt {5} \sqrt [3]{1-x^6}}-\frac {\left (5+\sqrt {5}\right ) \sqrt [3]{x^6-1} x^2 \operatorname {AppellF1}\left (\frac {1}{3},1,-\frac {1}{3},\frac {4}{3},\frac {2 x^6}{3+\sqrt {5}},x^6\right )}{10 \left (3+\sqrt {5}\right ) \sqrt [3]{1-x^6}}+\frac {\sqrt [3]{x^6-1} x^2 \operatorname {AppellF1}\left (\frac {1}{3},1,-\frac {1}{3},\frac {4}{3},\frac {2 x^6}{3+\sqrt {5}},x^6\right )}{\sqrt {5} \sqrt [3]{1-x^6}}+\frac {\sqrt [3]{x^6-1} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},-\frac {1}{6},\frac {5}{6},x^6\right )}{\sqrt [3]{1-x^6} x} \]