Logarithms, Trignometry, Integrals (Part 3)

Calculus Level 5

\[ \large \int_0^{\pi /2} \ln \left( \left | \cos(\tan x ) \right | \right) \, dx \]

The value of the integral above is equal to \[ \dfrac {A\pi }B \left( \ln\left( \dfrac{C+e^{-D}}F \right) \right) \]

where \(A,B,C,D\) and \(F\) are positive integers with \( A,B\) coprime and \(\displaystyle e = \lim_{n\to\infty} \left( 1 + \dfrac1n\right)^n \approx 2.71828 \).

Compute \(A+B+C+D+F\).

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