An interesting triple integral
Following Gerald Edgar,
$$W_3(1)\equiv\mathbb{E}[|Y|]=\int dx\int dy \,\sqrt{x^2+y^2}\,p(x,y)=2\pi\int_0^\infty r^2p(r)dr$$
with $p(x,y)dxdy=p(r)\,rdrd\phi$ the rotationally invariant distribution of the complex variable $Y=x+iy=r\cos\phi+ir\sin\phi$.
From here
$$\mathbb{E}[|X|]=\int dx\int dy \,|x|\,p(x,y)=\int_0^\infty \int_0^{2\pi} |\cos\phi|r^2 p(r)\,drd\phi=\frac{2}{\pi}W_3(1).$$
According to equation (12) of Borwein, Straub, and Wan the numbers $W_n(1)$ are given by
$$W_1(1)=1,\;\;W_n(1)=n\int_0^\infty J_1(x)J_0(x)^{n-1}\frac{dx}{x},\;\;n\geq 2.$$
We thus arrive at
$$\int\limits_0^{2\pi}\int\limits_0^{2\pi}\int\limits_0^{2\pi}|\cos x+\cos y+\cos z|\ dx\ dy\ dz=(2\pi)^3\mathbb{E}[|X|]=16\pi^2 W_3(1)$$
$$\qquad=48\pi^2\int_0^\infty J_1(x)J_0(x)^{2}\frac{dx}{x}=3\;\frac{2^{1/3}}{\pi^2}\Gamma({\textstyle \frac{1}{3}})^6
+
108\;\frac{2^{2/3}}{\pi^2}\Gamma({\textstyle \frac{2}{3}})^6 \approx 248.65$$
comments:
- since $(2\pi)^3\approx 248.05$, the average of the absolute value of the sum of three cosines equals unity within one-quarter of a percent.
- the $n$-fold integral $$I_n\equiv\int_0^{2\pi}\cdots\int_0^{2\pi}|\cos x_1+\cdots + \cos x_n|\,dx_1\cdots dx_n=2^{n+1}\pi^{n-1}n\int_0^\infty J_1(x)J_0(x)^{n-1}\frac{dx}{x}$$ equals a power of 2 for $n=1$ and $n=2$, but not for $n=3$ (nor for larger $n$, as far as I have checked).
- for $n\gg 1$ one can use the diffusion equation to deduce that
$$\lim_{n\rightarrow\infty}(\pi/n)^{1/2}(2\pi)^{-n}I_n=1.$$
(I have not been able to derive this asymptotics directly from the Bessel function expression for $I_n$.)
Comment
This is related to some papers (e.g. this) by P.M. Borwein et. al. on short random walks in the plane.
Let $X_1, X_2, \dots$ be i.i.d. random variables, uniformly distributed on the unit circle $|z|=1$ in the complex plane. Then $$ X_1+X_2+X_3 $$ is a random variable in the plane, and your integral is $$ (2\pi)^3\;\mathbb{E}\Big[\big|\mathrm{Re}(X_1+X_2+X_3)\big|\Big] $$
Borwein and collaborators have information on moments $$ W_3(s) := \mathbb{E}\Big[\big|X_1+X_2+X_3\big|^s\Big] $$ including "closed form" in terms of hypergeometric functions when $s \in \mathbb N$. In particular $$ W_3(1) = \frac{3}{16}\;\frac{2^{1/3}}{\pi^4}\Gamma({\textstyle \frac{1}{3}})^6 + \frac{27}{4}\;\frac{2^{2/3}}{\pi^4}\Gamma({\textstyle \frac{2}{3}})^6 . $$
Now, if the distribution of $$ Y = X_1+X_2+X_3 $$ is rotationally symmetric in the complex plane, and we have the exact value of $\mathbb E[|Y|]$, can we find $\mathbb E[|\mathrm{Re}\;Y|]$ ??