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