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