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 $a+b+c+d+f+g+h+j+k$ .