Cmment
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 $s \in \mathbb N$.

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 $\mathbb E[|\mathrm{Re}\;Y|]$ ??

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 $\mathbb E[|\mathrm{Re}\;Y|]$ ??