Hyperbolic sine times hyperbolic cosine

$\begin{aligned} I & = \int \sinh^4 x \cosh^5 x \ dx \\ & = \int \left(\frac {e^x - e^{-x}}2\right)^4 \left(\frac {e^x + e^{-x}}2\right)^5 dx & \small \color{#3D99F6} \text{Let }u = e^x \implies du = e^x dx \\ & = \frac 1{512} \int (u^2-u^{-2})^4(u+u^{-1}) \frac {du}u \\ & = \frac 1{512} \int \left(u^8 + u^6 - 4u^4 - 4u^2 + 6 + 6u^{-2} - 4u^{-4} - 4x^{-6} + u^{-8} + u^{-10}\right) du \\ & = \frac 1{512} \left(\frac {u^9}9 + \frac {u^7}7 - \frac {4u^5}5 - \frac {4u^3}3 +6u - 6u^{-1} + \frac {4u^{-3}}3 + \frac {4u^{-5}}5 - \frac {u^{-7}}7 - \frac {u^{-9}}9 \right) + \color{#3D99F6} K & \small \color{#3D99F6} \text{where } K \text{ is the constant of integration.} \\ & = \frac {3\sinh x}{128} - \frac {\sinh 3x}{192} - \frac {\sinh 5x}{320} + \frac {\sinh 7x}{1792} + \frac {\sinh 9x}{2304} + K \end{aligned}$

Therefore, $|A+B+C+D-E| = |3+192+1+1792-2304| = \boxed{316}$ .