\[ \int \frac {1728 x^2+864 e x^2-864 x^3+e^{2 x} \left (12 x^2+12 x^3-6 x^4+e \left (6 x^2+6 x^3\right )\right )+e^x \left (288 x^2+72 x^3-72 x^4+e \left (144 x^2+72 x^3\right )\right )+\left (e^x \left (216 x+72 e x-72 x^2\right )+e^{2 x} \left (18 x+6 e x-6 x^2\right )\right ) \log ^2(3+e-x)+\left (-432 x-144 e x+144 x^2+e^x \left (-36 x-12 e x+12 x^2\right )\right ) \log (x)+\left (-144 x-72 e x+72 x^2+e^x \left (-12 x-12 x^2+6 x^3+e \left (-6 x-6 x^2\right )\right )\right ) \log ^2(x)+(36+12 e-12 x) \log ^3(x)+\log (3+e-x) \left (1728 x+864 e x-864 x^2+e^{2 x} \left (12 x+30 x^2-12 x^3+e \left (6 x+12 x^2\right )\right )+e^x \left (288 x+288 x^2-144 x^3+e \left (144 x+144 x^2\right )\right )+\left (-432-144 e+144 x+e^x (-36-12 e+12 x)\right ) \log (x)+e^x \left (-18 x-6 e x+6 x^2\right ) \log ^2(x)\right )}{3 x+e x-x^2} \, dx \]
Optimal antiderivative \[ 3 \left (\ln \left (x \right )^{2}-\left (12+{\mathrm e}^{x}\right ) \left (\ln \left (3-x +{\mathrm e}\right )+x \right )\right )^{2} \]
command
integrate((((6*x*exp(1)-6*x**2+18*x)*exp(x)**2+(72*x*exp(1)-72*x**2+216*x)*exp(x))*ln(3-x+exp(1))**2+((-6*x*exp(1)+6*x**2-18*x)*exp(x)*ln(x)**2+((-12*exp(1)+12*x-36)*exp(x)-144*exp(1)+144*x-432)*ln(x)+((12*x**2+6*x)*exp(1)-12*x**3+30*x**2+12*x)*exp(x)**2+((144*x**2+144*x)*exp(1)-144*x**3+288*x**2+288*x)*exp(x)+864*x*exp(1)-864*x**2+1728*x)*ln(3-x+exp(1))+(12*exp(1)-12*x+36)*ln(x)**3+(((-6*x**2-6*x)*exp(1)+6*x**3-12*x**2-12*x)*exp(x)-72*x*exp(1)+72*x**2-144*x)*ln(x)**2+((-12*x*exp(1)+12*x**2-36*x)*exp(x)-144*x*exp(1)+144*x**2-432*x)*ln(x)+((6*x**3+6*x**2)*exp(1)-6*x**4+12*x**3+12*x**2)*exp(x)**2+((72*x**3+144*x**2)*exp(1)-72*x**4+72*x**3+288*x**2)*exp(x)+864*x**2*exp(1)-864*x**3+1728*x**2)/(x*exp(1)-x**2+3*x),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ 432 x^{2} - 72 x \log {\left (x \right )}^{2} + \left (864 x - 72 \log {\left (x \right )}^{2}\right ) \log {\left (- x + e + 3 \right )} + \left (3 x^{2} + 6 x \log {\left (- x + e + 3 \right )} + 3 \log {\left (- x + e + 3 \right )}^{2}\right ) e^{2 x} + \left (72 x^{2} - 6 x \log {\left (x \right )}^{2} + 144 x \log {\left (- x + e + 3 \right )} - 6 \log {\left (x \right )}^{2} \log {\left (- x + e + 3 \right )} + 72 \log {\left (- x + e + 3 \right )}^{2}\right ) e^{x} + 3 \log {\left (x \right )}^{4} + 432 \log {\left (- x + e + 3 \right )}^{2} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________