\[ \int \frac {e^{2 x} \left (4 x+16 x^2\right )+e^x \left (-8 x-32 x^2\right ) \log (3)+\left (4 x+16 x^2\right ) \log ^2(3)+\left (e^{2 x} \left (-8 x-16 x^2\right )-8 x^3 \log (3)+\left (-8 x-16 x^2\right ) \log ^2(3)+e^x \left (8 x^3+\left (16 x+32 x^2\right ) \log (3)\right )\right ) \log (x)-4 e^x x^4 \log ^2(x)}{\left (e^{2 x} \left (1+8 x+16 x^2\right )+e^x \left (-2-16 x-32 x^2\right ) \log (3)+\left (1+8 x+16 x^2\right ) \log ^2(3)\right ) \log ^2(x)+\left (e^x \left (2 x^2+8 x^3\right )+\left (-2 x^2-8 x^3\right ) \log (3)\right ) \log ^3(x)+x^4 \log ^4(x)} \, dx \]
Optimal antiderivative \[ \frac {4 x}{\ln \! \left (x \right ) \left (\frac {x \ln \left (x \right )}{\ln \left (3\right )-{\mathrm e}^{x}}-\frac {1+4 x}{x}\right )} \]
command
Integrate[(E^(2*x)*(4*x + 16*x^2) + E^x*(-8*x - 32*x^2)*Log[3] + (4*x + 16*x^2)*Log[3]^2 + (E^(2*x)*(-8*x - 16*x^2) - 8*x^3*Log[3] + (-8*x - 16*x^2)*Log[3]^2 + E^x*(8*x^3 + (16*x + 32*x^2)*Log[3]))*Log[x] - 4*E^x*x^4*Log[x]^2)/((E^(2*x)*(1 + 8*x + 16*x^2) + E^x*(-2 - 16*x - 32*x^2)*Log[3] + (1 + 8*x + 16*x^2)*Log[3]^2)*Log[x]^2 + (E^x*(2*x^2 + 8*x^3) + (-2*x^2 - 8*x^3)*Log[3])*Log[x]^3 + x^4*Log[x]^4),x]
Mathematica 13.1 output
\[ \int \frac {e^{2 x} \left (4 x+16 x^2\right )+e^x \left (-8 x-32 x^2\right ) \log (3)+\left (4 x+16 x^2\right ) \log ^2(3)+\left (e^{2 x} \left (-8 x-16 x^2\right )-8 x^3 \log (3)+\left (-8 x-16 x^2\right ) \log ^2(3)+e^x \left (8 x^3+\left (16 x+32 x^2\right ) \log (3)\right )\right ) \log (x)-4 e^x x^4 \log ^2(x)}{\left (e^{2 x} \left (1+8 x+16 x^2\right )+e^x \left (-2-16 x-32 x^2\right ) \log (3)+\left (1+8 x+16 x^2\right ) \log ^2(3)\right ) \log ^2(x)+\left (e^x \left (2 x^2+8 x^3\right )+\left (-2 x^2-8 x^3\right ) \log (3)\right ) \log ^3(x)+x^4 \log ^4(x)} \, dx \]
Mathematica 12.3 output
\[ -\frac {4 x^2 \left (\frac {e^{2 x} (1+4 x)+(1+4 x) \log ^2(3)-e^x (8 x \log (3)+\log (9))}{\log (x)}-\frac {x^2 \left (e^{3 x} \left (-2-11 x-8 x^2+16 x^3\right )+2 \log ^3(3)+12 x \log ^3(3)+x^2 \log ^2(3) (1+16 \log (3))+x^3 \left (-4 \log ^2(3)+\log (9) \log (81)\right )+e^x \left (-2 \log ^2(3)+4 x^3 \left (4 \log ^2(3)-\log (9)\right )-\log ^2(9)-x^2 \left (8 \log ^2(3)+\log (9)+8 \log (3) \log (81)\right )-x \left (11 \log ^2(3)+8 \log (3) \log (9)+\log (9) \log (81)\right )\right )+e^{2 x} \left (x^3 (4-32 \log (3))+x^2 (1+8 \log (81))+\log (729)+2 x \log (129140163)\right )\right )}{\left (x^2+e^x \left (-2-3 x+4 x^2\right )+\log (9)+x \log (81)\right ) \left ((1+4 x) \left (e^x-\log (3)\right )+x^2 \log (x)\right )}\right )}{(1+4 x)^2 \left (e^x-\log (3)\right )^2} \]