Multiparty Computation to calculate addition of shares without revealling individual shares
I have $n$ persons, each holding a secret integer $x_i$ ($i$ from $1$ to $n$) and I'm looking for a way for them to jointly compute the sum of these secrets without revealing to each other their individual secrets.

A simple application of arithmetic secret-sharing based secure multi-party computation (&quot;arithmetic GMW&quot;) can do that.
The protocol for that goes as follows:

(One-Time Setup) Pick a prime $p$ that is guaranteed to be larger than the sum of all input values (you can safely overshoot the sume by a large margin here, e.g. pick a prime larger than $2^{96}$ for 64-bit inputs). This prime will be used to define the finite field we will work in. If you already have a preferred one, use it here. All arithmetic that follows will be carried out in this field.
Have every party $P_i$ choose $n-1$ random integers uniformly at random from the finite field. Call them $s_{i,j}$ and let $s_{i,n}=x_i-\sum_{j=1}^{n-1}s_{i,j}$. This creates an arithmetic $n$-out-of-$n$ sharing of $x_i$
Have every party $P_i$ send $s_{i,j}$ to $P_{j}$.
Have every party $P_i$ compute $r_i=\sum_{\ell=1}^{n}s_{\ell,i}$, that is, have them all sum up all the shares they receive.
Have every party $P_i$ broadcast their result $r_i$ to all other parties (or only to the one needing to know the sum).
Have every party with with all $r_j$ compute $\sigma=\sum_{j=1}^{n}r_j$. This value is the desired sum.