\[ \int \frac {e^{\frac {-8+x-2 \log (x)+10 x \log (4-2 x-\log (5))}{2 x \log (4-2 x-\log (5))}} \left (8 x-x^2+(-12+6 x+3 \log (5)) \log (4-2 x-\log (5))+\log (x) (2 x+(-4+2 x+\log (5)) \log (4-2 x-\log (5)))\right )}{\left (-4 x^2+2 x^3+x^2 \log (5)\right ) \log ^2(4-2 x-\log (5))} \, dx \]
Optimal antiderivative \[ {\mathrm e}^{4+\frac {\frac {\frac {x}{2}-\ln \left (x \right )-4}{\ln \left (-\ln \left (5\right )+4-2 x \right )}+x}{x}} \]
command
integrate((((ln(5)+2*x-4)*ln(-ln(5)+4-2*x)+2*x)*ln(x)+(3*ln(5)+6*x-12)*ln(-ln(5)+4-2*x)-x**2+8*x)*exp(1/2*(-2*ln(x)+10*x*ln(-ln(5)+4-2*x)-8+x)/x/ln(-ln(5)+4-2*x))/(x**2*ln(5)+2*x**3-4*x**2)/ln(-ln(5)+4-2*x)**2,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Exception raised: TypeError} \]
Sympy 1.8 under Python 3.8.8 output
\[ e^{\frac {5 x \log {\left (- 2 x - \log {\left (5 \right )} + 4 \right )} + \frac {x}{2} - \log {\left (x \right )} - 4}{x \log {\left (- 2 x - \log {\left (5 \right )} + 4 \right )}}} \]