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Complex integration, where the integrand is an infinite sum with respect to a distribution

I am trying to solve the following complex integration problem:
$$\int\limits_{0}^{\infty}\left(e^z-e^{ -z}-ze^{ -z}\right)x^ze^{ -x}dx$$
Since the integrand is not elementary, I tried to separate its real and imaginary parts, then integrate over the real and imaginary parts separately. Doing so, I get:
$$\int\limits_{0}^{\infty}\left(e^{x-i\pi x}+e^{x+i\pi x}\right)x^xe^{ -x}dx+i\int\limits_{0}^{\infty}\left(e^{ -x+i\pi x}+e^{ -x-i\pi x}\right)x^xe^{ -x}dx$$
Now, the problem is that I cannot deal with the following terms:
$$\left(e^{x-i\pi x}+e^{x+i\pi x}\right)$$
and
$$\left(e^{ -x+i\pi x}+e^{ -x-i\pi x}\right)$$
Because if I try to take the real part of these expressions, for example, I get a hypergeometric expression where I have to take the Euler gamma function. And the gamma function is not elementary.
There is some subtlety here. The integrand is
$(f_0(x)+if_1(x))(g_0(x)+ig_1(x))$, where $f_0(x)$ is some function which I deal with and $g_0(x)$ and $g_

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