\[ \int \frac {\left (1+x^3\right ) \sqrt {-1+x^6}}{x^{13} \left (-1+x^3\right )} \, dx \]
Optimal antiderivative \[ \frac {\sqrt {x^{6}-1}\, \left (32 x^{9}+21 x^{6}+16 x^{3}+6\right )}{72 x^{12}}+\frac {7 \arctan \left (x^{3}+\sqrt {x^{6}-1}\right )}{12} \]
command
Int[((1 + x^3)*Sqrt[-1 + x^6])/(x^13*(-1 + x^3)),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \frac {7 \sqrt {x^6-1} \arctan \left (\sqrt {x^3-1} \sqrt {x^3+1}\right )}{24 \sqrt {x^3-1} \sqrt {x^3+1}}+\frac {7 \sqrt {x^6-1}}{24 x^6}+\frac {\sqrt {x^6-1}}{12 x^{12}}+\frac {2 \sqrt {x^6-1}}{9 x^9}+\frac {4 \sqrt {x^6-1}}{9 x^3} \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \int \frac {\left (1+x^3\right ) \sqrt {-1+x^6}}{x^{13} \left (-1+x^3\right )} \, dx \]________________________________________________________________________________________