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