Smooth Approximation of $\sin^2\left({\frac {n \pi}{\lfloor{x}\rfloor}}\right)$ and Its Indefinite Integral

Question

What is a smooth function that approximates $\sin^2\left({\dfrac {n \pi}{\lfloor{x}\rfloor}}\right)$ (it must at least have maxima and minima only at the integers and at the integers be equal to $\sin^2\left({\dfrac {n \pi}{x}}\right)$? And what is its integral (it can use special functions, like $\mathrm{Si}$ or $\mathrm{Ci}$, but not $\Gamma$)?