Trigonometry and Calculus!

Calculus Level 5

\[\large F(x)=\dfrac{\sin x\cos x}{\sqrt{19\sin^2 x+10\cos^2 x+6}}\]

Let \(0< \alpha < \dfrac \pi 2 \) such that \(\max(F(x)) = F(\alpha) = \phi \).

Given that the value of \[\displaystyle \large \mathcal H = \int_{\pi (9\phi - 1)}^\alpha F(x) \, dx \] is equal to \( \dfrac AB(\sqrt C - D) \), where \(A,B,C\) and \(D\) are positive integers with \(A,B\) coprime and \(C\) square-free.

Find \(A+B+C+D\).

Inspired by a problem that was written by Shivam Jadhav.

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