\[\large{\lim_{n \to \infty} \int_{\pi/4}^{\pi/3} n \ln{\left[\left(1+\frac{\sin\theta\sec^2\theta}{n}\right)^{\cos\theta} \left(1+\frac{\cos\theta}{n}\right)^{\cot\theta} \left(1+\frac{\cot\theta}{n}\right)^{\sin\theta\sec^2\theta}\right]}d\theta}\]

Suppose the value of the above limit can be expressed in the form: \[\ln\left({\frac{(\sqrt{A} + 1)^B}{\sqrt{C}}}\right) +\left(\frac{1}{D} - \frac{1}{\sqrt{E}}\right)\]

where \(A,B,C,D,E\) are positive integers, and \(A,C,E\) are square free. Then what is the value of \(A+B+C+D+E \) ?

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