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