\[ \int \frac {\sqrt [4]{x^2+x^4}}{x^4 \left (-1+x^4\right )} \, dx \]
Optimal antiderivative \[ \frac {2 \left (x^{2}+1\right ) \left (x^{4}+x^{2}\right )^{\frac {1}{4}}}{5 x^{3}}+\frac {\arctan \left (\frac {2^{\frac {1}{4}} x}{\left (x^{4}+x^{2}\right )^{\frac {1}{4}}}\right ) 2^{\frac {1}{4}}}{2}-\frac {\arctanh \left (\frac {2^{\frac {1}{4}} x}{\left (x^{4}+x^{2}\right )^{\frac {1}{4}}}\right ) 2^{\frac {1}{4}}}{2} \]
command
Integrate[(x^2 + x^4)^(1/4)/(x^4*(-1 + x^4)),x]
Mathematica 13.1 output
\[ \frac {\sqrt [4]{x^2+x^4} \left (4 \left (1+x^2\right )^{5/4}+5 \sqrt [4]{2} x^{5/2} \text {ArcTan}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^2}}\right )-5 \sqrt [4]{2} x^{5/2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^2}}\right )\right )}{10 x^3 \sqrt [4]{1+x^2}} \]
Mathematica 12.3 output
\[ \int \frac {\sqrt [4]{x^2+x^4}}{x^4 \left (-1+x^4\right )} \, dx \]________________________________________________________________________________________