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