Si $I = \int_{e^{\pi/6}}^{e^{\pi/2}} \frac{\sin(\ln(\sin(\ln x)))\cos(\ln x)}{x \sin(\ln x)} dx$, entonces el valor de $\cos^{-1}(I+1)$ es igual a:

$$ \begin{array}{llll} \text{(a) } \frac{\pi}{4} & \text{(b) } \frac{\pi}{3} & \text{(c) } \ln 2 & \text{(d) } 2 \ln 2 \end{array} $$