\[ \int \frac {\left (-1+x^3\right )^3 \left (1+x^3\right ) \sqrt {2+3 x^6+2 x^{12}}}{x^7 \left (1+x^6\right )} \, dx \]
Optimal antiderivative \[ \frac {\left (x^{6}-4 x^{3}+1\right ) \sqrt {2 x^{12}+3 x^{6}+2}}{6 x^{6}}-\frac {4 \arctan \left (\frac {x^{3}}{\sqrt {2}+\sqrt {2}\, x^{6}+\sqrt {2 x^{12}+3 x^{6}+2}}\right )}{3}+\frac {\ln \left (x \right ) \sqrt {2}}{4}-\frac {\ln \left (\sqrt {2}+\sqrt {2}\, x^{6}+\sqrt {2 x^{12}+3 x^{6}+2}\right ) \sqrt {2}}{12} \]
command
Integrate[((-1 + x^3)^3*(1 + x^3)*Sqrt[2 + 3*x^6 + 2*x^12])/(x^7*(1 + x^6)),x]
Mathematica 13.1 output
\[ \frac {1}{12} \left (\frac {2 \left (1-4 x^3+x^6\right ) \sqrt {2+3 x^6+2 x^{12}}}{x^6}-16 \text {ArcTan}\left (\frac {x^3}{\sqrt {2}+\sqrt {2} x^6+\sqrt {2+3 x^6+2 x^{12}}}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1+\frac {3 x^6}{2}+x^{12}}}{1+x^6}\right )\right ) \]
Mathematica 12.3 output
\[ \int \frac {\left (-1+x^3\right )^3 \left (1+x^3\right ) \sqrt {2+3 x^6+2 x^{12}}}{x^7 \left (1+x^6\right )} \, dx \]________________________________________________________________________________________