\( D \) is the solid inside the cylinder \( x^{2}+y^{2}=9 \), between the planes \( z=7 \) and \( z=x+17 \). Use the Divergence Theorem to find the flux of the vector field \( \mathbf{F}(x, y, z)=\left\langle x y-\sqrt{y^{2}+z^{2}}, 5 y+e^{z^{3}}, 3 z-x^{2} \cot y\right\rangle \) over the surface of \( D \). Consider cylindrical coordinates. Write the exact answer. Do not round.

Step 1 ：Define cylindrical coordinates transformation: \((x, y, z) = (r \cos \theta, r \sin \theta, z)\) with \(0 \le r \le 3\), \(0 \le \theta \le 2 \pi\), \(7 \le z \le r \cos \theta + 17 \).

Step 2 ：Find the divergence of \( \mathbf{F} \): \(\operatorname{div}(\mathbf{F}) = \nabla \cdot \mathbf{F} = \frac{\partial}{\partial x}(xy-\sqrt{y^{2}+z^{2}}) + \frac{\partial}{\partial y}(5y+e^{z^{3}}) + \frac{\partial}{\partial z}(3z-x^{2}\cot y)\).

Step 3 ：Calculate the triple integral using Divergence Theorem: \( \iiint_D \operatorname{div}(\mathbf{F}) \, dV = \int_0^{2\pi} \int_0^3 \int_7^{r\cos\theta + 17} \operatorname{div}(\mathbf{F}) \, r \, dz \, dr \, d\theta \).