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