Problem
I'm attempting to decompose a system prepared in state $|\Psi_\mu\rangle$, defined by
$$\Psi_\mu(x) = \left( \frac{m\omega}{\pi \hbar} \right)^{1/4} \exp \left( -\frac{m\omega (x-\mu)^2}{2\hbar} \right)$$
where $H_n$ is the nth Hermite polynomial
$$H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n}e^{-x^2}= \frac{d^n}{dt^n}e^{2xt-t^2}\Big\vert_{t=0} \, ,$$
into eigenstates of the quantum harmonic oscillator Hamiltonian.
My Approach
I know that any arbitrary state, such as $\Psi_\mu(x)$, can be expressed as a superposition of eigenstates of the Hamiltonian $\psi_n(x)$, because they form a complete, orthonormal basis
$$\Psi_\mu(x) = \sum\limits_{n=0}^{\infty}c_n \psi_n(x)$$
with
$$\psi_n(x) = \sqrt{\frac{1}{2^n n!}} \left( \frac{m \omega}{\pi \hbar} \right)^{1/4} H_n \left( \sqrt{\frac{m\omega}{\hbar}} x \right) \exp \left( -\frac{m\omega x^2}{2\hbar} \right) $$
with the weights $c_n$ given by
$$c_n = \int dx \psi_n^*(x) \Psi_\mu(x) \, .$$
So, if I can solve this integral for index $n$, I have decomposed $\Psi_\mu(x)$.
My problem
I'm not certain how to evaluate the integral for $c_n$
\begin{align} c_n =& \sqrt{\frac{1}{2^n n!}} \left( \frac{m \omega}{\pi \hbar} \right)^{1/2}\\ & \int dx H_n \left( \sqrt{\frac{m\omega}{\hbar}}x \right) \exp \left( -\frac{m\omega x^2}{2\hbar} \right) \exp \left( -\frac{m\omega (x-\mu)^2}{2\hbar} \right) \, . \end{align}
Having to integrate around the nth derivative of x in $H_n(x)$ is throwing me off. I feel like I'm either unaware of some existing machinery for integrating Hermite polynomials (although my search has mainly turned up results dealing with orthnormality) or there is another, more elegant approach.