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