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