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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