Weirdo Integral

Calculus Level 5

Consider the following integral \[\begin{equation} {\huge\int}_{\large 0}^{\Large \pi}\sqrt{\frac{1}{2}\sqrt{\frac{1}{2}\sqrt{\frac{1}{2}\sqrt{\frac{1}{2}\sqrt{\frac{1+\cos x}{2}}+\frac{1}{2}}+\frac{1}{2}}+\frac{1}{2}}+\frac{1}{2}}\,\,\,\,\,{\large dx} \end{equation}\] If the closed-form of the above integral can be expressed as \(\displaystyle a^4\sqrt{b-\sqrt{c+\sqrt{d+\sqrt{e}}}}\), then find \(\displaystyle abcde\).

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