Too many parts! (My fourteenth integral problem)

Calculus Level 5

\[ \large \displaystyle \int_{0}^{\pi} e^{x} x^{3} \cos x \, dx \]

If the above integral can be expressed in the form \[ -\dfrac{a}{b} - \dfrac{c e^{\pi}}{d} + \dfrac{f\pi e^{\pi}}{g} - \dfrac{h \pi^{j} e^{\pi}}{k}, \] where \(a, b, c, d, f, g, h, j\) and \( k \) are positive integers with \(\gcd(a, b) = \gcd(c, d) = \gcd(f, g) = \gcd (h, k) = 1 \), and \( e \) is Euler's number, find \( a+b+c+d+f+g+h+j+k\).

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