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