Numerical verification of the estimate:

How to verify numerically with considerable accuracy in Mathematica the following :

$$int_2^xdfrac{1}{zGamma(sin^2[πGamma(z)/(2z)])}dzsimln(ln(x))$$

?

I need more suitable and better code for it as you can see integrand oscillates wildly.

Any help is appreciated

I removed my code as I know it isn’t going to help after x =10.