Let $a_n={\int }_{-1}^n\left(1+\frac{x}{2}+\frac{x^2}{3}+\dots +\frac{x^{n-1}}{n}\right)dx$ for every $n\in N$. Then the sum of all the elements of the set $\left\{n\in \mathrm{N}:a_n\in \left(2,30\right)\right\}$ is _________.

Given $a_n={\int }_{-1}^n\left(1+\frac{x}{2}+\frac{x^2}{3}+\dots .+\frac{x^{n-1}}{n}\right)dx$

$={\left[x+\frac{x^2}{2^2}+\frac{x^3}{3^2}+\dots \dots +\frac{x^n}{n^2}\right]}_{-1}^n$

i.e. $a_n=\frac{n+1}{1^2}+\frac{n^2-1}{2^2}+\frac{n^3+1}{3^2}+\frac{n^4-1}{4^2}+\dots +\frac{n^n+{\left(-1\right)}^{n+1}}{n^2}$

Here $a_1=2, a_2=\frac{2+1}{1}+\frac{2^2-1}{2}=\frac{9}{2},$

$a_3=4+2+\frac{28}{9}=\frac{100}{9},$

$a_4=5+\frac{15}{4}+\frac{65}{9}+\frac{255}{16}>31.$

$\because a_n\in \left(2,30\right)$, so the required set is $\left\{2,3\right\}$.

$\therefore$  sum of all elements $=5$.