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