\[ \int \frac {e^2 (-1-2 x)+6 x+14 x^2+4 x^3+\left (e^2-6 x-2 x^2\right ) \log \left (\frac {-e^2+6 x+2 x^2}{2 x}\right )+\log (x) \left (-3 e^2+12 x+2 x^2+\left (e^2-6 x-2 x^2\right ) \log \left (\frac {-e^2+6 x+2 x^2}{2 x}\right )\right )}{e^2-6 x-2 x^2} \, dx \]
Optimal antiderivative \[ x -x^{2}-1+\ln \left (x \right ) x \left (\ln \left (x -\frac {{\mathrm e}^{2}}{2 x}+3\right )-2\right ) \]
command
integrate((((exp(2)-2*x**2-6*x)*ln(1/2*(-exp(2)+2*x**2+6*x)/x)-3*exp(2)+2*x**2+12*x)*ln(x)+(exp(2)-2*x**2-6*x)*ln(1/2*(-exp(2)+2*x**2+6*x)/x)+(-1-2*x)*exp(2)+4*x**3+14*x**2+6*x)/(exp(2)-2*x**2-6*x),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ - x^{2} - 2 x \log {\left (x \right )} + x + \left (x \log {\left (x \right )} + \frac {5}{48}\right ) \log {\left (\frac {x^{2} + 3 x - \frac {e^{2}}{2}}{x} \right )} + \frac {5 \log {\left (x \right )}}{48} - \frac {5 \log {\left (x^{2} + 3 x - \frac {e^{2}}{2} \right )}}{48} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Exception raised: CoercionFailed} \]________________________________________________________________________________________