\[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1+x^8} \, dx \]
Optimal antiderivative \[ \frac {\arctan \left (\frac {2^{\frac {1}{8}} x}{\left (x^{6}+x^{2}\right )^{\frac {1}{4}}}\right ) 2^{\frac {5}{8}}}{4}+\frac {\arctan \left (\frac {2^{\frac {5}{8}} x \left (x^{6}+x^{2}\right )^{\frac {1}{4}}}{x^{2} 2^{\frac {1}{4}}-\sqrt {x^{6}+x^{2}}}\right ) 2^{\frac {1}{8}}}{4}-\frac {\arctanh \left (\frac {2^{\frac {1}{8}} x}{\left (x^{6}+x^{2}\right )^{\frac {1}{4}}}\right ) 2^{\frac {5}{8}}}{4}+\frac {\arctanh \left (\frac {\frac {x^{2} 2^{\frac {5}{8}}}{2}+\frac {\sqrt {x^{6}+x^{2}}\, 2^{\frac {3}{8}}}{2}}{x \left (x^{6}+x^{2}\right )^{\frac {1}{4}}}\right ) 2^{\frac {1}{8}}}{4} \]
command
Integrate[((-1 + x^4)*(x^2 + x^6)^(1/4))/(1 + x^8),x]
Mathematica 13.1 output
\[ \frac {\sqrt [4]{x^2+x^6} \left (\sqrt {2} \text {ArcTan}\left (\frac {\sqrt [8]{2} \sqrt {x}}{\sqrt [4]{1+x^4}}\right )+\text {ArcTan}\left (\frac {2^{5/8} \sqrt {x} \sqrt [4]{1+x^4}}{\sqrt [4]{2} x-\sqrt {1+x^4}}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt [8]{2} \sqrt {x}}{\sqrt [4]{1+x^4}}\right )+\tanh ^{-1}\left (\frac {2\ 2^{3/8} \sqrt {x} \sqrt [4]{1+x^4}}{2 x+2^{3/4} \sqrt {1+x^4}}\right )\right )}{2\ 2^{7/8} \sqrt {x} \sqrt [4]{1+x^4}} \]
Mathematica 12.3 output
\[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1+x^8} \, dx \]________________________________________________________________________________________