\[ \int \frac {(4+3 x) \left (-1-x+x^4\right ) \sqrt [4]{-1-x+2 x^4}}{x^6 \left (1+x+x^4\right )} \, dx \]
Optimal antiderivative \[ -\frac {4 \left (2 x^{4}-x -1\right )^{\frac {1}{4}} \left (12 x^{4}-x -1\right )}{5 x^{5}}-4 \,3^{\frac {1}{4}} \arctan \left (\frac {3^{\frac {1}{4}} x}{\left (2 x^{4}-x -1\right )^{\frac {1}{4}}}\right )+4 \,3^{\frac {1}{4}} \arctanh \left (\frac {3^{\frac {1}{4}} x}{\left (2 x^{4}-x -1\right )^{\frac {1}{4}}}\right ) \]
command
Integrate[((4 + 3*x)*(-1 - x + x^4)*(-1 - x + 2*x^4)^(1/4))/(x^6*(1 + x + x^4)),x]
Mathematica 13.1 output
\[ -\frac {4 \sqrt [4]{-1-x+2 x^4} \left (-1-x+12 x^4\right )}{5 x^5}-4 \sqrt [4]{3} \text {ArcTan}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{-1-x+2 x^4}}\right )+4 \sqrt [4]{3} \tanh ^{-1}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{-1-x+2 x^4}}\right ) \]
Mathematica 12.3 output
\[ \int \frac {(4+3 x) \left (-1-x+x^4\right ) \sqrt [4]{-1-x+2 x^4}}{x^6 \left (1+x+x^4\right )} \, dx \]________________________________________________________________________________________