\[ \int \frac {e^{\frac {5+20 x^2-5 \log \left (\frac {4+e^{2 x}}{5 x^2}\right )}{x}} \left (20+80 x^2+e^{2 x} \left (5-10 x+20 x^2\right )+\left (20+5 e^{2 x}\right ) \log \left (\frac {4+e^{2 x}}{5 x^2}\right )\right )}{4 x^2+e^{2 x} x^2} \, dx \]
Optimal antiderivative \[ {\mathrm e}^{\frac {5+20 x^{2}-5 \ln \left (\frac {4+{\mathrm e}^{2 x}}{5 x^{2}}\right )}{x}} \]
command
Integrate[(E^((5 + 20*x^2 - 5*Log[(4 + E^(2*x))/(5*x^2)])/x)*(20 + 80*x^2 + E^(2*x)*(5 - 10*x + 20*x^2) + (20 + 5*E^(2*x))*Log[(4 + E^(2*x))/(5*x^2)]))/(4*x^2 + E^(2*x)*x^2),x]
Mathematica 13.1 output
\[ 5^{5/x} e^{\frac {5}{x}+20 x} \left (\frac {4+e^{2 x}}{x^2}\right )^{-5/x} \]
Mathematica 12.3 output
\[ \int \frac {e^{\frac {5+20 x^2-5 \log \left (\frac {4+e^{2 x}}{5 x^2}\right )}{x}} \left (20+80 x^2+e^{2 x} \left (5-10 x+20 x^2\right )+\left (20+5 e^{2 x}\right ) \log \left (\frac {4+e^{2 x}}{5 x^2}\right )\right )}{4 x^2+e^{2 x} x^2} \, dx \]________________________________________________________________________________________