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