My thirteenth integration problem

Calculus Level 5

\[ \large \displaystyle \int_{0}^{\pi} x^3 \sin \left( \dfrac{\pi}{4} -x \right) \sin (x) \sin \left( \dfrac{\pi}{4} + x \right) \cos (x) \, dx \]

If the above definite integral can be expressed in the form \[ \dfrac{a \pi}{b} - \dfrac{c \pi^{d}}{e} , \] where \(a,b,c,d\) and \(e\) are all positive integers and \( \gcd(a, b) = \gcd(c, e) = 1 \), find \(a+b+c+d+e\).

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