Integral involving Elliptic Integral

Calculus Level 5

The following integral evaluates to:

01(E(k)π2)dkk=a+πb+c0π/4ln(cosx)dx\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 aa, bb and cc are integers. Find a+b+c|a|+|b|+|c|.

Details and Assumptions:

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