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