\[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1-x^4+x^8} \, dx \]
Optimal antiderivative \[ \frac {\arctan \left (\frac {3^{\frac {1}{8}} x}{\left (x^{6}+x^{2}\right )^{\frac {1}{4}}}\right ) 3^{\frac {5}{8}}}{6}-\frac {\arctan \left (\frac {\sqrt {2}\, 3^{\frac {7}{8}} x \left (x^{6}+x^{2}\right )^{\frac {1}{4}}}{-3 x^{2}+3^{\frac {3}{4}} \sqrt {x^{6}+x^{2}}}\right ) \sqrt {2}\, 3^{\frac {5}{8}}}{12}-\frac {\arctanh \left (\frac {3^{\frac {1}{8}} x}{\left (x^{6}+x^{2}\right )^{\frac {1}{4}}}\right ) 3^{\frac {5}{8}}}{6}+\frac {\arctanh \left (\frac {\frac {3^{\frac {1}{8}} x^{2} \sqrt {2}}{2}+\frac {\sqrt {x^{6}+x^{2}}\, \sqrt {2}\, 3^{\frac {7}{8}}}{6}}{x \left (x^{6}+x^{2}\right )^{\frac {1}{4}}}\right ) \sqrt {2}\, 3^{\frac {5}{8}}}{12} \]
command
Integrate[((-1 + x^4)*(x^2 + x^6)^(1/4))/(1 - x^4 + x^8),x]
Mathematica 13.1 output
\[ \frac {\sqrt [4]{x^2+x^6} \left (2 \text {ArcTan}\left (\frac {\sqrt [8]{3} \sqrt {x}}{\sqrt [4]{1+x^4}}\right )+\sqrt {2} \text {ArcTan}\left (\frac {\sqrt {2} 3^{7/8} \sqrt {x} \sqrt [4]{1+x^4}}{3 x-3^{3/4} \sqrt {1+x^4}}\right )-2 \tanh ^{-1}\left (\frac {\sqrt [8]{3} \sqrt {x}}{\sqrt [4]{1+x^4}}\right )+\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} 3^{7/8} \sqrt {x} \sqrt [4]{1+x^4}}{3 x+3^{3/4} \sqrt {1+x^4}}\right )\right )}{4\ 3^{3/8} \sqrt {x} \sqrt [4]{1+x^4}} \]
Mathematica 12.3 output
\[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1-x^4+x^8} \, dx \]________________________________________________________________________________________