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The dirac equation with some small gauge potential $\epsilon \gamma^\mu{A}_\mu(x)$ reads as $$(\gamma^\mu\partial_\mu-m+\epsilon\gamma^\mu A_\mu(x))\psi(x) = 0.$$

The solution up to first order is

$$ \psi(x) = \psi_0(x) +\tau\int\frac{d^4p}{(2\pi)^4}\int d^4x'\frac{e^{-ip(x-x')}}{\gamma_\mu{p}^\mu-m} \gamma^\mu A_\mu(x')\psi_0(x')+\mathcal{O}(\tau^2).$$

How to solve this integral for the coulomb field?

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There seems to be a problem with your question. You need to define all components of $A_\mu(x)$, so it is not clear so far what you are asking. Another thing. Even if we ignore the first problem, the expression for $A(x)$ does not depend on $x$, so the electromagnetic field vanishes. If this is indeed what you are interested in, you can use solutions for the free Dirac equation and apply a gauge transform.

EDIT (2/23/19): Your edited question also has its share of problems. What is $x$ in your definition of $A(x)$? If it is a 4-vector, then the definition does not make sense. If, however, it is the $x$ coordinate, then your $A(x)$ is a gradient of a scalar function, so the electromagnetic field vanishes, and you can use gauge-transformed solutions of the free Dirac equation.

EDIT (2/23/19): I would like to note that there are exact solutions of the Dirac equation both for the Coulomb field and the uniform electric field (F. Sauter, http://www.neo-classical-physics.info/uploads/3/4/3/6/34363841/sauter_-_electron_in_homo_electric_field.pdf).

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