Evaluating $\int_{B^n(1)}e^{-|\vec{x}|}\frac{\cos|\vec{x}|}{\left(\sum_{m=2}^{n}x_{m}^2\right)^{(n-2)/2}}\,\mathrm{d}^nx$

Question

$$ \int_{B^n(1)}e^{-|\vec{x}|}\frac{\cos|\vec{x}|}{\left(\sum_{m=2}^{n}x_{m}^2\right)^{(n-2)/2}}\,\mathrm{d}^nx $$

The problem is to calculate this integral and express the solution as Beta and/or Gamma functions, and the domain of integral is the $n$-dimensional ball with radius $1$.