I would like to calculate this $k$-th moment:
$$\int_{-\infty}^{+\infty} \quad x^k\quad \left(i^n\frac{\sin(\pi a x+\frac{n\pi}{2})}{\pi ax+\frac{n\pi}{2}}+(-i)^n\frac{\sin(\pi a x-\frac{n\pi}{2})}{\pi ax-\frac{n\pi}{2}}\right)^2\quad \exp(-bx^2) \,dx$$
with $k\geq0$ and $n\in \mathbf{N}^+$.
Thanks