\[ \int \frac {\left (-2+x^2\right ) \left (-1+x^2\right ) \sqrt [4]{-1+x^2+x^4}}{x^6 \left (-1+x^2+2 x^4\right )} \, dx \]
Optimal antiderivative \[ \frac {2 \left (x^{4}+x^{2}-1\right )^{\frac {1}{4}} \left (9 x^{4}-x^{2}+1\right )}{5 x^{5}}+\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x \left (x^{4}+x^{2}-1\right )^{\frac {1}{4}}}{-x^{2}+\sqrt {x^{4}+x^{2}-1}}\right )-\sqrt {2}\, \arctanh \left (\frac {\sqrt {2}\, x \left (x^{4}+x^{2}-1\right )^{\frac {1}{4}}}{x^{2}+\sqrt {x^{4}+x^{2}-1}}\right ) \]
command
Integrate[((-2 + x^2)*(-1 + x^2)*(-1 + x^2 + x^4)^(1/4))/(x^6*(-1 + x^2 + 2*x^4)),x]
Mathematica 13.1 output
\[ \frac {2 \sqrt [4]{-1+x^2+x^4} \left (1-x^2+9 x^4\right )}{5 x^5}+\sqrt {2} \text {ArcTan}\left (\frac {\sqrt {2} x \sqrt [4]{-1+x^2+x^4}}{-x^2+\sqrt {-1+x^2+x^4}}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{-1+x^2+x^4}}{x^2+\sqrt {-1+x^2+x^4}}\right ) \]
Mathematica 12.3 output
\[ \int \frac {\left (-2+x^2\right ) \left (-1+x^2\right ) \sqrt [4]{-1+x^2+x^4}}{x^6 \left (-1+x^2+2 x^4\right )} \, dx \]________________________________________________________________________________________