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