I'm interested in finding a nice representation of $$ \int_{-j}^{n-1-j} \mathrm{sinc}(\pi x)\exp(-ikx) \mathrm{d}x $$ where $n,j$ are integers with $0 \le j < n$.
Maple gets nothing, and Wolfram alpha says "standard computation time exceeded".
I can do it numerically (of course), but another representation would be cool to have; asymptotics and other approximations are welcome.
As @reuns wrote in comments, the antiderivative is quite simple if you accept special functions (the exponential integral function).
Consider $$I=\int \frac {e^{i \pi x}} x e^{-i k x} \,dx=\int \frac{e^{i (\pi -k) x}}{x}\,dx=\text{Ei}(i (\pi -k) x)$$ $$J=\int \frac {e^{-i \pi x}} x e^{-i k x} \,dx=\int \frac{e^{-i (\pi+k ) x}}{x}\,dx=\text{Ei}(-i (\pi+k ) x)$$ which make $$\int \frac{\sin(\pi x)}x e^{-i k x} \,dx=\frac{i }{2 \pi }\big(\text{Ei}(-i (\pi+k ) x)-\text{Ei}(i (\pi -k) x)\big)$$ $$\int \frac{\cos(\pi x)}x e^{-i k x} \,dx=\frac{1}{2} \big(\text{Ei}(-i (\pi+k ) x)+\text{Ei}(i (\pi -k) x)\big)$$