Integral involving Elliptic Integral

Calculus Level 5

The following integral evaluates to:

\[\int_0^1 \left(E(k)-\dfrac{\pi}{2}\right)\,\frac{dk}{k}=a+\dfrac{\pi}{b}+c\int_0^{\pi/4} \ln(\cos x)\,dx\]

where \(a\), \(b\) and \(c\) are integers. Find \(|a|+|b|+|c|\).

Details and Assumptions:

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