I'm interested in $\text{Ci}(2\pi n)$ for integers $n\geq 1$.
As the graph below shows, as $n$ increases the cosine integral seems to (strictly?) monotonically decrease. I've looked online but can't find much, and I'm wondering - is there a closed form expression for such values?
I have surmised that
$$\lim_{n\to\infty}(2\pi n)^2\text{Ci}(2\pi n) = -1.$$
I did not find any closed form but here is an expansion for large values of $n$ $$\text{Ci}(2 \pi n)=-\frac{1}{4 \pi ^2 n^2}+\frac{3}{8 \pi ^4 n^4}-\frac{15}{8 \pi ^6 n^6}+\frac{315}{16 \pi ^8 n^8}+O\left(\left(\frac{1}{n}\right)^9\right)$$ Hoping that this helps a little.
What should be of interest to you is to plot $n^3 \text{Ci}(2 \pi n)$ as a function of $n$ even for small values : it looks like a straight line going through origin.
You could find interesting also,for large values of $n$, $$\text{Si}(2 \pi n)=\frac{\pi }{2}-\frac{1}{2 \pi n}+\frac{1}{4 \pi ^3 n^3}-\frac{3}{4 \pi ^5 n^5}+\frac{45}{8 \pi ^7 n^7}+O\left(\left(\frac{1}{n}\right)^{8}\right)$$