\[ \int \frac {1+x^6}{\sqrt [4]{x^3+x^5} \left (1-x^6\right )} \, dx \]
Optimal antiderivative \[ \frac {2 \arctan \left (\frac {x}{\left (x^{5}+x^{3}\right )^{\frac {1}{4}}}\right )}{3}+\frac {\arctan \left (\frac {2^{\frac {1}{4}} x}{\left (x^{5}+x^{3}\right )^{\frac {1}{4}}}\right ) 2^{\frac {3}{4}}}{6}-\frac {\arctan \left (\frac {2^{\frac {3}{4}} x \left (x^{5}+x^{3}\right )^{\frac {1}{4}}}{x^{2} \sqrt {2}-\sqrt {x^{5}+x^{3}}}\right ) 2^{\frac {1}{4}}}{6}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x \left (x^{5}+x^{3}\right )^{\frac {1}{4}}}{-x^{2}+\sqrt {x^{5}+x^{3}}}\right )}{3}+\frac {2 \arctanh \left (\frac {x}{\left (x^{5}+x^{3}\right )^{\frac {1}{4}}}\right )}{3}+\frac {\arctanh \left (\frac {2^{\frac {1}{4}} x}{\left (x^{5}+x^{3}\right )^{\frac {1}{4}}}\right ) 2^{\frac {3}{4}}}{6}+\frac {\arctanh \left (\frac {\frac {x^{2} 2^{\frac {3}{4}}}{2}+\frac {\sqrt {x^{5}+x^{3}}\, 2^{\frac {1}{4}}}{2}}{x \left (x^{5}+x^{3}\right )^{\frac {1}{4}}}\right ) 2^{\frac {1}{4}}}{6}+\frac {\sqrt {2}\, \arctanh \left (\frac {\frac {x^{2} \sqrt {2}}{2}+\frac {\sqrt {x^{5}+x^{3}}\, \sqrt {2}}{2}}{x \left (x^{5}+x^{3}\right )^{\frac {1}{4}}}\right )}{3} \]
command
Integrate[(1 + x^6)/((x^3 + x^5)^(1/4)*(1 - x^6)),x]
Mathematica 13.1 output
\[ \frac {x^{3/4} \sqrt [4]{1+x^2} \left (4 \text {ArcTan}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x^2}}\right )+2^{3/4} \text {ArcTan}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x^2}}\right )-\sqrt [4]{2} \text {ArcTan}\left (\frac {2^{3/4} \sqrt [4]{x} \sqrt [4]{1+x^2}}{\sqrt {2} \sqrt {x}-\sqrt {1+x^2}}\right )+2 \sqrt {2} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{x} \sqrt [4]{1+x^2}}{-\sqrt {x}+\sqrt {1+x^2}}\right )+4 \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x^2}}\right )+2^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x^2}}\right )+2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{x} \sqrt [4]{1+x^2}}{\sqrt {x}+\sqrt {1+x^2}}\right )+\sqrt [4]{2} \tanh ^{-1}\left (\frac {2 \sqrt [4]{2} \sqrt [4]{x} \sqrt [4]{1+x^2}}{2 \sqrt {x}+\sqrt {2} \sqrt {1+x^2}}\right )\right )}{6 \sqrt [4]{x^3+x^5}} \]
Mathematica 12.3 output
\[ \int \frac {1+x^6}{\sqrt [4]{x^3+x^5} \left (1-x^6\right )} \, dx \]________________________________________________________________________________________