\[ \int \frac {1-x^4}{\left (1+x^2+x^4\right ) \sqrt [4]{-x^3+x^5}} \, dx \]
Optimal antiderivative \[ \frac {\arctan \left (\frac {3^{\frac {7}{8}} \sqrt {2-\sqrt {2}}\, x \left (x^{5}-x^{3}\right )^{\frac {1}{4}}}{-3 x^{2}+3^{\frac {3}{4}} \sqrt {x^{5}-x^{3}}}\right ) 2^{\frac {1}{4}}}{2 \left (51+36 \sqrt {2}\right )^{\frac {1}{8}}}+\frac {\left (\frac {17}{3}+4 \sqrt {2}\right )^{\frac {1}{8}} \arctan \left (\frac {3^{\frac {7}{8}} \sqrt {2+\sqrt {2}}\, x \left (x^{5}-x^{3}\right )^{\frac {1}{4}}}{-3 x^{2}+3^{\frac {3}{4}} \sqrt {x^{5}-x^{3}}}\right ) 2^{\frac {1}{4}}}{2}+\frac {\arctanh \left (\frac {\frac {3^{\frac {1}{8}} x^{2}}{\sqrt {2-\sqrt {2}}}+\frac {\sqrt {x^{5}-x^{3}}\, 3^{\frac {7}{8}}}{3 \sqrt {2-\sqrt {2}}}}{x \left (x^{5}-x^{3}\right )^{\frac {1}{4}}}\right ) 2^{\frac {1}{4}}}{2 \left (51+36 \sqrt {2}\right )^{\frac {1}{8}}}+\frac {\left (\frac {17}{3}+4 \sqrt {2}\right )^{\frac {1}{8}} \arctanh \left (\frac {\frac {3^{\frac {1}{8}} x^{2}}{\sqrt {2+\sqrt {2}}}+\frac {\sqrt {x^{5}-x^{3}}\, 3^{\frac {7}{8}}}{3 \sqrt {2+\sqrt {2}}}}{x \left (x^{5}-x^{3}\right )^{\frac {1}{4}}}\right ) 2^{\frac {1}{4}}}{2} \]
command
Integrate[(1 - x^4)/((1 + x^2 + x^4)*(-x^3 + x^5)^(1/4)),x]
Mathematica 13.1 output
\[ \frac {x^{3/4} \sqrt [4]{-1+x^2} \left (\text {ArcTan}\left (\frac {\sqrt [4]{2} 3^{7/8} \sqrt [4]{x} \sqrt [4]{-1+x^2}}{\sqrt [8]{17+12 \sqrt {2}} \left (-3 \sqrt {x}+3^{3/4} \sqrt {-1+x^2}\right )}\right )+\sqrt [4]{17+12 \sqrt {2}} \text {ArcTan}\left (\frac {\sqrt [4]{2} 3^{7/8} \sqrt [8]{17+12 \sqrt {2}} \sqrt [4]{x} \sqrt [4]{-1+x^2}}{-3 \sqrt {x}+3^{3/4} \sqrt {-1+x^2}}\right )+\tanh ^{-1}\left (\frac {\sqrt [4]{2} 3^{7/8} \sqrt [4]{x} \sqrt [4]{-1+x^2}}{\sqrt [8]{17+12 \sqrt {2}} \left (3 \sqrt {x}+3^{3/4} \sqrt {-1+x^2}\right )}\right )+\sqrt [4]{17+12 \sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt [4]{2} 3^{7/8} \sqrt [8]{17+12 \sqrt {2}} \sqrt [4]{x} \sqrt [4]{-1+x^2}}{3 \sqrt {x}+3^{3/4} \sqrt {-1+x^2}}\right )\right )}{2^{3/4} \sqrt [8]{51+36 \sqrt {2}} \sqrt [4]{x^3 \left (-1+x^2\right )}} \]
Mathematica 12.3 output
\[ \int \frac {1-x^4}{\left (1+x^2+x^4\right ) \sqrt [4]{-x^3+x^5}} \, dx \]________________________________________________________________________________________