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