\[ \int \frac {x^6 \left (4+x^3\right )}{\left (1+x^3\right )^{3/4} \left (-1-2 x^3-x^6+x^8\right )} \, dx \]
Optimal antiderivative \[ \arctan \left (\frac {x}{\left (x^{3}+1\right )^{\frac {1}{4}}}\right )-\frac {\sqrt {2}\, \arctan \left (\frac {-\frac {x^{2} \sqrt {2}}{2}+\frac {\sqrt {x^{3}+1}\, \sqrt {2}}{2}}{x \left (x^{3}+1\right )^{\frac {1}{4}}}\right )}{2}-\arctanh \left (\frac {x}{\left (x^{3}+1\right )^{\frac {1}{4}}}\right )-\frac {\sqrt {2}\, \arctanh \left (\frac {\sqrt {2}\, x \left (x^{3}+1\right )^{\frac {1}{4}}}{x^{2}+\sqrt {x^{3}+1}}\right )}{2} \]
command
Integrate[(x^6*(4 + x^3))/((1 + x^3)^(3/4)*(-1 - 2*x^3 - x^6 + x^8)),x]
Mathematica 13.1 output
\[ \text {ArcTan}\left (\frac {x}{\sqrt [4]{1+x^3}}\right )-\frac {\text {ArcTan}\left (\frac {-x^2+\sqrt {1+x^3}}{\sqrt {2} x \sqrt [4]{1+x^3}}\right )}{\sqrt {2}}-\tanh ^{-1}\left (\frac {x}{\sqrt [4]{1+x^3}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{1+x^3}}{x^2+\sqrt {1+x^3}}\right )}{\sqrt {2}} \]
Mathematica 12.3 output
\[ \int \frac {x^6 \left (4+x^3\right )}{\left (1+x^3\right )^{3/4} \left (-1-2 x^3-x^6+x^8\right )} \, dx \]________________________________________________________________________________________