\[ \int \frac {e^{-\frac {1250}{x \log (2 x)}} (-2500-2500 \log (2 x))+e^{-\frac {2500}{x \log (2 x)}} (2500+2500 \log (2 x))}{x^2 \log ^2(2 x)} \, dx \]
Optimal antiderivative \[ \left (1-{\mathrm e}^{-\frac {1250}{x \ln \left (2 x \right )}}\right )^{2} \]
command
integrate(((2500*ln(2*x)+2500)*exp(-1250/x/ln(2*x))**2+(-2500*ln(2*x)-2500)*exp(-1250/x/ln(2*x)))/x**2/ln(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
\[ - 2 e^{- \frac {1250}{x \log {\left (2 x \right )}}} + e^{- \frac {2500}{x \log {\left (2 x \right )}}} \]