Complete Elliptic Integral Of The First Kind?

Calculus Level 5

\[\dfrac{1}{2}\int_{-\pi/2}^{\pi/2}\dfrac{d\theta}{\sqrt{1-(\sqrt{2}-1)^2\sin^2{\theta}}}=\dfrac{\sqrt{\sqrt{A}+B}}{A^{C/D}\sqrt{\pi}}\Gamma\left(\dfrac{B}{E}\right)\Gamma\left(\dfrac{F}{E}\right)\]

The equation above holds true positive integers \(A,B,C,D,E\) and \(F\) such that \(A\) is square-free and \(C,D\) are coprime. Find \(A+B+C+D+E+F\).

Notation: \( \Gamma(\cdot) \) denotes the Gamma function.

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