Gaussian clusters and original distributions

In Gaussian clustering (i.e. General Mixture Models) we model the data with some clusters. For example, in the below figure, we have two clusters $C_1, C_2$, each of which are modeled with a Gaussian (Normal) distribution: $p_{C_1} \sim \mathcal{N}(\mu_1, \Sigma_1)$ and $p_{C_2} \sim \mathcal{N}(\mu_2, \Sigma_2)$ with proportions $\pi_1$ and $\pi_2$. The probability that a given point $x$ belongs to $C_1$ (is generated from $C_1$) is: $$p(C_1 | x ) = \frac { \pi_1 \mathcal{N}(\mu_1, \Sigma_1)} {\sum_{i=1}^2{ \pi_i \mathcal{N}(\mu_i, \Sigma_i) } }$$ The probability that $x$ belongs to $C_2$ can be similarly calculated.

Now, assume that we have a group of points that are generated (sampled) from another Gaussian distribution $q(x) \sim \mathcal{N}(\mu_q, \Sigma_q)$. Obviously, we can compute the probability of each point belonging to each cluster. However, we instead want to compute the expected probability of a sample belonging to each cluster. In other words, if we generate infinite samples from the $q(x)$, what is the average (expected) value of $p(C_1 | x )$ for all points?

**P.S. ** I need the solution in closed-form.

$$\int p(x|C_i) q(x)\, dx$$ but wouldn't you rather have $P(C_i | x)$ and the expected values of these, i.e. probabilities of classification?