Is there a simple formula for: \begin{equation} \sum_{s,j,k,l=0}^N\int_{\mathbb{R}} \frac{cos(st)sin(jt)cos(kt)sin(lt)}{2jl} dt, \end{equation} where $N \in \mathbb{N}$?
Number Theory-esque Conclusion:
Unless I made a typo I reduced it to: \begin{align} & \sum_{s,j,k,l=0}^N\int_{\mathbb{R}} \frac{ \left( sin((i+j) t) + sin((i-j) t)\right)* \left( sin((k+l) t) + sin((k-l) t)\right) }{4jk} dt \\ & = \frac{\sum_{s,j,k,l=0}^N \delta_{i+j = k+l} +\delta_{i+j = k-l} +\delta_{i-j = k+l} + \delta_{i-j = k-l} }{4jk} \\ & = \frac{\sum_{s,j,k,l=0}^N \delta_{i+j = k-l} }{2jk} + \frac{\sum_{s,j,k,l=0}^N \delta_{i+j = k+l} + \delta_{i-j = k-l} }{4jk} \end{align}, but even this seems very difficult as it reduces to a counting argument which I'm uncertain of how to approach.