\[ \int \frac {\left (-32 x^2+16 x^3+e^2 \left (8 x-6 x^2\right )+\left (-16 x^2+4 x^3+e^2 \left (8 x-2 x^2\right )\right ) \log \left (32 x^2-16 x^3+2 x^4+e^2 \left (-16 x+8 x^2-x^3\right )\right )\right ) \log \left (\frac {5}{x \log \left (32 x^2-16 x^3+2 x^4+e^2 \left (-16 x+8 x^2-x^3\right )\right )}\right )+\left (16 x^2-4 x^3+e^2 \left (-8 x+2 x^2\right )\right ) \log \left (32 x^2-16 x^3+2 x^4+e^2 \left (-16 x+8 x^2-x^3\right )\right ) \log ^2\left (\frac {5}{x \log \left (32 x^2-16 x^3+2 x^4+e^2 \left (-16 x+8 x^2-x^3\right )\right )}\right )}{\left (e^2 (-4+x)+8 x-2 x^2\right ) \log \left (32 x^2-16 x^3+2 x^4+e^2 \left (-16 x+8 x^2-x^3\right )\right )} \, dx \]
Optimal antiderivative \[ x^{2} \ln \left (\frac {5}{x \ln \left (x \left (4-x \right )^{2} \left (2 x -{\mathrm e}^{2}\right )\right )}\right )^{2} \]
command
integrate((((2*x**2-8*x)*exp(2)-4*x**3+16*x**2)*ln((-x**3+8*x**2-16*x)*exp(2)+2*x**4-16*x**3+32*x**2)*ln(5/x/ln((-x**3+8*x**2-16*x)*exp(2)+2*x**4-16*x**3+32*x**2))**2+(((-2*x**2+8*x)*exp(2)+4*x**3-16*x**2)*ln((-x**3+8*x**2-16*x)*exp(2)+2*x**4-16*x**3+32*x**2)+(-6*x**2+8*x)*exp(2)+16*x**3-32*x**2)*ln(5/x/ln((-x**3+8*x**2-16*x)*exp(2)+2*x**4-16*x**3+32*x**2)))/((x-4)*exp(2)-2*x**2+8*x)/ln((-x**3+8*x**2-16*x)*exp(2)+2*x**4-16*x**3+32*x**2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ x^{2} \log {\left (\frac {5}{x \log {\left (2 x^{4} - 16 x^{3} + 32 x^{2} + \left (- x^{3} + 8 x^{2} - 16 x\right ) e^{2} \right )}} \right )}^{2} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Exception raised: CoercionFailed} \]________________________________________________________________________________________