Weirdo Integral

Calculus Level 5

Consider the following integral 0π121212121+cosx2+12+12+12+12dx {\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}

If the closed-form of the above integral can be expressed as a4bc+d+e\displaystyle a^4\sqrt{b-\sqrt{c+\sqrt{d+\sqrt{e}}}}, then find abcde\displaystyle abcde.

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