\[ \int \frac {1+x^4}{\left (1-x^4\right ) \sqrt [4]{x^3+x^5}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (x^{5}+x^{3}\right )^{\frac {3}{4}}}{x^{2} \left (x^{2}+1\right )}+\frac {\arctan \left (\frac {2^{\frac {1}{4}} x}{\left (x^{5}+x^{3}\right )^{\frac {1}{4}}}\right ) 2^{\frac {3}{4}}}{4}-\frac {\arctan \left (\frac {2^{\frac {3}{4}} x \left (x^{5}+x^{3}\right )^{\frac {1}{4}}}{x^{2} \sqrt {2}-\sqrt {x^{5}+x^{3}}}\right ) 2^{\frac {1}{4}}}{4}+\frac {\arctanh \left (\frac {2^{\frac {1}{4}} x}{\left (x^{5}+x^{3}\right )^{\frac {1}{4}}}\right ) 2^{\frac {3}{4}}}{4}+\frac {\arctanh \left (\frac {\frac {x^{2} 2^{\frac {3}{4}}}{2}+\frac {\sqrt {x^{5}+x^{3}}\, 2^{\frac {1}{4}}}{2}}{x \left (x^{5}+x^{3}\right )^{\frac {1}{4}}}\right ) 2^{\frac {1}{4}}}{4} \]
command
Integrate[(1 + x^4)/((1 - x^4)*(x^3 + x^5)^(1/4)),x]
Mathematica 13.1 output
\[ \frac {x^{3/4} \left (8 \sqrt [4]{x}+2^{3/4} \sqrt [4]{1+x^2} \text {ArcTan}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x^2}}\right )-\sqrt [4]{2} \sqrt [4]{1+x^2} \text {ArcTan}\left (\frac {2^{3/4} \sqrt [4]{x} \sqrt [4]{1+x^2}}{\sqrt {2} \sqrt {x}-\sqrt {1+x^2}}\right )+2^{3/4} \sqrt [4]{1+x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x^2}}\right )+\sqrt [4]{2} \sqrt [4]{1+x^2} \tanh ^{-1}\left (\frac {2 \sqrt [4]{2} \sqrt [4]{x} \sqrt [4]{1+x^2}}{2 \sqrt {x}+\sqrt {2} \sqrt {1+x^2}}\right )\right )}{4 \sqrt [4]{x^3+x^5}} \]
Mathematica 12.3 output
\[ \int \frac {1+x^4}{\left (1-x^4\right ) \sqrt [4]{x^3+x^5}} \, dx \]________________________________________________________________________________________