Trigonometric Integration

Calculus Level 4

\[\displaystyle\int_0^{\pi/2}\int_0^{\pi/2}\left(\sqrt{1+\cos(x-y)+\cos(x+y)\cos(x-y)+\cos(x+y)}\right)\mathrm{d}x~\mathrm{d}y\]

If the above expression can be simplified to \(\large \color{red}{\alpha}\pi^{\color{purple}{\large\psi}}\) for some \(\large\color{red}{\alpha},\color{purple}{\psi}\in\mathbb{Z}\),then:

\[\Large(\color{red}{\alpha}+\color{purple}{\psi}+6)^{(\color{red}{\alpha}-\color{purple}{\psi})!}=\ ?\]

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