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