\[ \int \frac {4-2 x+e^2 \left (2 x-x^2\right )+e^{\frac {x^2+\log ^2\left (\frac {2+e^2 x}{x}\right )}{x}} \left (-2 x-4 x^2+e^2 \left (-x^2-2 x^3\right )+8 \log \left (\frac {2+e^2 x}{x}\right )+\left (4+2 e^2 x\right ) \log ^2\left (\frac {2+e^2 x}{x}\right )\right )}{16+24 x+12 x^2+2 x^3+e^2 \left (8 x+12 x^2+6 x^3+x^4\right )+e^{\frac {3 \left (x^2+\log ^2\left (\frac {2+e^2 x}{x}\right )\right )}{x}} \left (2 x^3+e^2 x^4\right )+e^{\frac {2 \left (x^2+\log ^2\left (\frac {2+e^2 x}{x}\right )\right )}{x}} \left (12 x^2+6 x^3+e^2 \left (6 x^3+3 x^4\right )\right )+e^{\frac {x^2+\log ^2\left (\frac {2+e^2 x}{x}\right )}{x}} \left (24 x+24 x^2+6 x^3+e^2 \left (12 x^2+12 x^3+3 x^4\right )\right )} \, dx \]
Optimal antiderivative \[ \frac {x}{\left (x +{\mathrm e}^{x +\frac {\ln \left (\frac {{\mathrm e}^{2} x +2}{x}\right )^{2}}{x}} x +2\right )^{2}} \]
command
integrate((((2*exp(2)*x+4)*log((exp(2)*x+2)/x)^2+8*log((exp(2)*x+2)/x)+(-2*x^3-x^2)*exp(2)-4*x^2-2*x)*exp((log((exp(2)*x+2)/x)^2+x^2)/x)+(-x^2+2*x)*exp(2)+4-2*x)/((x^4*exp(2)+2*x^3)*exp((log((exp(2)*x+2)/x)^2+x^2)/x)^3+((3*x^4+6*x^3)*exp(2)+6*x^3+12*x^2)*exp((log((exp(2)*x+2)/x)^2+x^2)/x)^2+((3*x^4+12*x^3+12*x^2)*exp(2)+6*x^3+24*x^2+24*x)*exp((log((exp(2)*x+2)/x)^2+x^2)/x)+(x^4+6*x^3+12*x^2+8*x)*exp(2)+2*x^3+12*x^2+24*x+16),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {x}{x^{2} e^{\left (\frac {2 \, {\left (x^{2} + \log \left (\frac {x e^{2} + 2}{x}\right )^{2}\right )}}{x}\right )} + 2 \, x^{2} e^{\left (\frac {x^{2} + \log \left (\frac {x e^{2} + 2}{x}\right )^{2}}{x}\right )} + x^{2} + 4 \, x e^{\left (\frac {x^{2} + \log \left (\frac {x e^{2} + 2}{x}\right )^{2}}{x}\right )} + 4 \, x + 4} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________