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