While solving Real Integrals, do you obtain results in Complex Numbers?

Calculus Level 5

$\large{ \int_{\pi/4}^{\pi/3} \left(\frac{1}{\tan x\ (\ln( \sin x))}+\frac{\tan x}{\ln (\cos x)}\right)\ \mathrm{d}x }$

If the above integral equals to $$z$$, $$z \in \mathbb C$$, where $$\mathbb C$$ denotes the set of complex numbers, evaluate the value of $$|z|$$ upto three correct places of decimals.

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