I need to find the solution of this integral:

$$\int^\pi_{-\pi}{d\varphi\,{}\frac{\Gamma(n,2\pi\varphi)}{(1+ia\varphi)^n}},\tag{1}\label{1}$$

where $a\in(0,1)$ and $n$ is a positive integer (not zero). The solution should take a form as closed as possible, i.e., not contain a sum or another integral. Since neither Mathematica nor Maple offers a direct solution, I've been able to reduce the problem to solving either of the two following expressions:

\begin{gather*} i\int_{-2\pi}^{2\pi}dt\,{}e^{-t}\left(\frac{1}{t}+\frac{ia}{2}\right)^{1-n},\tag{2}\label{2} \\ i^{n+1}\sum_{k=0}^\infty\frac{(-a)^k\Gamma(n+k)}{2^k\Gamma(k+2)}\left[\Gamma(n+k+1,2i\pi)-\Gamma(n+k+1,-2i\pi)\right].\tag{3}\label{3} \end{gather*}

All three expressions are real and numerically can be seen to converge. The gamma function with two arguments is the incomplete gamma function, defined as

$$\Gamma(a,z)\equiv\int_{z}^\infty dt\,{} t^{a-1}e^{-t}.$$

A solution to any of \eqref{1}, \eqref{2}, or \eqref{3} depending only on $a$ and $n$ would be absolutely great, as well as a definitive argument on why they cannot be written in terms of the known functions (elementary, hypergeometric, etc).