To solve this integral problem, we need to analyze the given integral expression:

$\int_0^{\pi/4} \frac{\ln(\cot x)}{(\sin x)^{2009} + (\cos x)^{2009}} \cdot (\sin 2x)^{2008} \, dx = \frac{a^b \ln a}{c^2}$

where $a$, $b$, and $c$ are integers in their lowest form. The goal is to find the value of $a + b + c$.

Step-by-Step Solution:

1. Simplify $\sin 2x$:

   • Recall that $\sin 2x = 2 \sin x \cos x$.
   • Therefore, $(\sin 2x)^{2008} = (2 \sin x \cos x)^{2008} = 2^{2008} (\sin x)^{2008} (\cos x)^{2008}$.

2. **Rewrite