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