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