\[ \int \frac {e^{\frac {5+\left (-2-36 x^2-e^{2 x} x^2-24 x^3-4 x^4+e^x \left (12 x^2+4 x^3\right )\right ) \log \left (\frac {1}{2} \left (4+e^x\right )\right )}{\log \left (\frac {1}{2} \left (4+e^x\right )\right )}} \left (-5 e^x+\left (-288 x-288 x^2-64 x^3+e^{3 x} \left (-2 x-2 x^2\right )+e^x \left (24 x+24 x^2\right )+e^{2 x} \left (16 x+16 x^2+4 x^3\right )\right ) \log ^2\left (\frac {1}{2} \left (4+e^x\right )\right )\right )}{\left (4+e^x\right ) \log ^2\left (\frac {1}{2} \left (4+e^x\right )\right )} \, dx \]
Optimal antiderivative \[ {\mathrm e}^{\frac {5}{\ln \left (\frac {{\mathrm e}^{x}}{2}+2\right )}-\left (2 x +6-{\mathrm e}^{x}\right )^{2} x^{2}-2} \]
command
Int[(E^((5 + (-2 - 36*x^2 - E^(2*x)*x^2 - 24*x^3 - 4*x^4 + E^x*(12*x^2 + 4*x^3))*Log[(4 + E^x)/2])/Log[(4 + E^x)/2])*(-5*E^x + (-288*x - 288*x^2 - 64*x^3 + E^(3*x)*(-2*x - 2*x^2) + E^x*(24*x + 24*x^2) + E^(2*x)*(16*x + 16*x^2 + 4*x^3))*Log[(4 + E^x)/2]^2))/((4 + E^x)*Log[(4 + E^x)/2]^2),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {\exp \left (\frac {5+\left (-2-36 x^2-e^{2 x} x^2-24 x^3-4 x^4+e^x \left (12 x^2+4 x^3\right )\right ) \log \left (\frac {1}{2} \left (4+e^x\right )\right )}{\log \left (\frac {1}{2} \left (4+e^x\right )\right )}\right ) \left (-5 e^x+\left (-288 x-288 x^2-64 x^3+e^{3 x} \left (-2 x-2 x^2\right )+e^x \left (24 x+24 x^2\right )+e^{2 x} \left (16 x+16 x^2+4 x^3\right )\right ) \log ^2\left (\frac {1}{2} \left (4+e^x\right )\right )\right )}{\left (4+e^x\right ) \log ^2\left (\frac {1}{2} \left (4+e^x\right )\right )} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \exp \left (-4 x^4-4 \left (6-e^x\right ) x^3-\left (6-e^x\right )^2 x^2+\frac {5}{\log \left (\frac {1}{2} \left (e^x+4\right )\right )}-2\right ) \]