\[ \int \frac {e^{e^{\frac {2 \left (4+x \log \left (2 x+4 x^2+2 x^3\right )\right )}{\log \left (2 x+4 x^2+2 x^3\right )}}+4 e^{\frac {4+x \log \left (2 x+4 x^2+2 x^3\right )}{\log \left (2 x+4 x^2+2 x^3\right )}} x+4 x^2} \left (\left (8 x^2+8 x^3\right ) \log ^2\left (2 x+4 x^2+2 x^3\right )+e^{\frac {2 \left (4+x \log \left (2 x+4 x^2+2 x^3\right )\right )}{\log \left (2 x+4 x^2+2 x^3\right )}} \left (-8-24 x+\left (2 x+2 x^2\right ) \log ^2\left (2 x+4 x^2+2 x^3\right )\right )+e^{\frac {4+x \log \left (2 x+4 x^2+2 x^3\right )}{\log \left (2 x+4 x^2+2 x^3\right )}} \left (-16 x-48 x^2+\left (4 x+8 x^2+4 x^3\right ) \log ^2\left (2 x+4 x^2+2 x^3\right )\right )\right )}{\left (x+x^2\right ) \log ^2\left (2 x+4 x^2+2 x^3\right )} \, dx \]
Optimal antiderivative \[ {\mathrm e}^{\left (2 x +{\mathrm e}^{\frac {4}{\ln \left (\left (x^{2}+x \right ) \left (2 x +2\right )\right )}+x}\right )^{2}} \]
command
integrate((((2*x^2+2*x)*log(2*x^3+4*x^2+2*x)^2-24*x-8)*exp((x*log(2*x^3+4*x^2+2*x)+4)/log(2*x^3+4*x^2+2*x))^2+((4*x^3+8*x^2+4*x)*log(2*x^3+4*x^2+2*x)^2-48*x^2-16*x)*exp((x*log(2*x^3+4*x^2+2*x)+4)/log(2*x^3+4*x^2+2*x))+(8*x^3+8*x^2)*log(2*x^3+4*x^2+2*x)^2)*exp(exp((x*log(2*x^3+4*x^2+2*x)+4)/log(2*x^3+4*x^2+2*x))^2+4*x*exp((x*log(2*x^3+4*x^2+2*x)+4)/log(2*x^3+4*x^2+2*x))+4*x^2)/(x^2+x)/log(2*x^3+4*x^2+2*x)^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ e^{\left (4 \, x^{2} + 4 \, x e^{\left (x + \frac {4}{\log \left (2 \, x^{3} + 4 \, x^{2} + 2 \, x\right )}\right )} + e^{\left (2 \, x + \frac {8}{\log \left (2 \, x^{3} + 4 \, x^{2} + 2 \, x\right )}\right )}\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________