\[ \int \frac {1-x^4}{\left (1+x^4\right ) \sqrt [4]{x^3+x^5}} \, dx \]
Optimal antiderivative \[ \frac {\arctan \left (\frac {2^{\frac {1}{8}} x}{\left (x^{5}+x^{3}\right )^{\frac {1}{4}}}\right ) 2^{\frac {7}{8}}}{2}-\frac {\arctan \left (\frac {2^{\frac {5}{8}} x \left (x^{5}+x^{3}\right )^{\frac {1}{4}}}{x^{2} 2^{\frac {1}{4}}-\sqrt {x^{5}+x^{3}}}\right ) 2^{\frac {3}{8}}}{2}+\frac {\arctanh \left (\frac {2^{\frac {1}{8}} x}{\left (x^{5}+x^{3}\right )^{\frac {1}{4}}}\right ) 2^{\frac {7}{8}}}{2}+\frac {\arctanh \left (\frac {\frac {x^{2} 2^{\frac {5}{8}}}{2}+\frac {\sqrt {x^{5}+x^{3}}\, 2^{\frac {3}{8}}}{2}}{x \left (x^{5}+x^{3}\right )^{\frac {1}{4}}}\right ) 2^{\frac {3}{8}}}{2} \]
command
Integrate[(1 - x^4)/((1 + x^4)*(x^3 + x^5)^(1/4)),x]
Mathematica 13.1 output
\[ \frac {x^{3/4} \sqrt [4]{1+x^2} \left (\sqrt {2} \text {ArcTan}\left (\frac {\sqrt [8]{2} \sqrt [4]{x}}{\sqrt [4]{1+x^2}}\right )-\text {ArcTan}\left (\frac {2^{5/8} \sqrt [4]{x} \sqrt [4]{1+x^2}}{\sqrt [4]{2} \sqrt {x}-\sqrt {1+x^2}}\right )+\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt [8]{2} \sqrt [4]{x}}{\sqrt [4]{1+x^2}}\right )+\tanh ^{-1}\left (\frac {2\ 2^{3/8} \sqrt [4]{x} \sqrt [4]{1+x^2}}{2 \sqrt {x}+2^{3/4} \sqrt {1+x^2}}\right )\right )}{2^{5/8} \sqrt [4]{x^3+x^5}} \]
Mathematica 12.3 output
\[ \int \frac {1-x^4}{\left (1+x^4\right ) \sqrt [4]{x^3+x^5}} \, dx \]________________________________________________________________________________________