\[ \int \frac {\left (\frac {x}{\log \left (\frac {3}{4} \log \left (\frac {x}{2}\right )\right )}\right )^{\frac {1}{-1+e^2}} \left (-1+\log \left (\frac {x}{2}\right ) \log \left (\frac {3}{4} \log \left (\frac {x}{2}\right )\right )\right )}{\left (-x+e^2 x\right ) \log \left (\frac {x}{2}\right ) \log \left (\frac {3}{4} \log \left (\frac {x}{2}\right )\right )} \, dx \]
Optimal antiderivative \[ {\mathrm e}^{\frac {\ln \left (\frac {x}{\ln \left (\frac {3 \ln \left (\frac {x}{2}\right )}{4}\right )}\right )}{{\mathrm e}^{2}-1}} \]
command
integrate((ln(1/2*x)*ln(3/4*ln(1/2*x))-1)*exp(ln(x/ln(3/4*ln(1/2*x)))/(exp(2)-1))/(exp(2)*x-x)/ln(1/2*x)/ln(3/4*ln(1/2*x)),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \left (\frac {x}{\log {\left (\frac {3 \log {\left (\frac {x}{2} \right )}}{4} \right )}}\right )^{\frac {1}{-1 + e^{2}}} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________