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