\[ \int \frac {24 e^{4+2 x}+8 e^x x^2+\left (4 e^x x^2+e^{2 x} \left (12 e^4 x+21 x^2-3 e x^2\right )\right ) \log \left (\frac {e^{-2 x} \left (16 x^2+e^x \left (96 e^4 x+168 x^2-24 e x^2\right )+e^{2 x} \left (144 e^8+441 x^2-126 e x^2+9 e^2 x^2+e^4 (504 x-72 e x)\right )\right )}{9 x^2}\right )}{\left (4 x^2+e^x \left (12 e^4 x+21 x^2-3 e x^2\right )\right ) \log ^2\left (\frac {e^{-2 x} \left (16 x^2+e^x \left (96 e^4 x+168 x^2-24 e x^2\right )+e^{2 x} \left (144 e^8+441 x^2-126 e x^2+9 e^2 x^2+e^4 (504 x-72 e x)\right )\right )}{9 x^2}\right )} \, dx \]
Optimal antiderivative \[ \frac {{\mathrm e}^{x}}{\ln \! \left ({\left (\frac {\frac {4 x \,{\mathrm e}^{-x}}{3}+4 x +4 \,{\mathrm e}^{4}}{x}+3-{\mathrm e}\right )}^{2}\right )} \]
command
Int[(24*E^(4 + 2*x) + 8*E^x*x^2 + (4*E^x*x^2 + E^(2*x)*(12*E^4*x + 21*x^2 - 3*E*x^2))*Log[(16*x^2 + E^x*(96*E^4*x + 168*x^2 - 24*E*x^2) + E^(2*x)*(144*E^8 + 441*x^2 - 126*E*x^2 + 9*E^2*x^2 + E^4*(504*x - 72*E*x)))/(9*E^(2*x)*x^2)])/((4*x^2 + E^x*(12*E^4*x + 21*x^2 - 3*E*x^2))*Log[(16*x^2 + E^x*(96*E^4*x + 168*x^2 - 24*E*x^2) + E^(2*x)*(144*E^8 + 441*x^2 - 126*E*x^2 + 9*E^2*x^2 + E^4*(504*x - 72*E*x)))/(9*E^(2*x)*x^2)]^2),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {24 e^{4+2 x}+8 e^x x^2+\left (4 e^x x^2+e^{2 x} \left (12 e^4 x+21 x^2-3 e x^2\right )\right ) \log \left (\frac {e^{-2 x} \left (16 x^2+e^x \left (96 e^4 x+168 x^2-24 e x^2\right )+e^{2 x} \left (144 e^8+441 x^2-126 e x^2+9 e^2 x^2+e^4 (504 x-72 e x)\right )\right )}{9 x^2}\right )}{\left (4 x^2+e^x \left (12 e^4 x+21 x^2-3 e x^2\right )\right ) \log ^2\left (\frac {e^{-2 x} \left (16 x^2+e^x \left (96 e^4 x+168 x^2-24 e x^2\right )+e^{2 x} \left (144 e^8+441 x^2-126 e x^2+9 e^2 x^2+e^4 (504 x-72 e x)\right )\right )}{9 x^2}\right )} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \frac {e^x \left (21 e^x x-3 e^{x+1} x+4 x+12 e^{x+4}\right ) \log \left (\frac {e^{-2 x} \left (21 e^x x-3 e^{x+1} x+4 x+12 e^{x+4}\right )^2}{9 x^2}\right )}{\left (3 (7-e) e^x x+4 x+12 e^{x+4}\right ) \log ^2\left (\frac {e^{-2 x} \left (3 (7-e) e^x x+4 x+12 e^{x+4}\right )^2}{9 x^2}\right )} \]