# $\int \frac{1}{\sqrt{1 – x^2}} \text{exp}\left(-\frac{1}{2} \frac{a^2 + b x}{1 – x^2}\right) dx$

I tried several online integral solvers in-vain. Any directions?
$$\int \frac{1}{\sqrt{1 – x^2}} \text{exp}\left(-\frac{1}{2} \frac{a^2 + b x}{1 – x^2}\right) dx$$

Note that the quantity inside the integral is coming from the following:
$$\int\mathbb{E}[\delta(x_1) \delta(x_2)] d\rho = \int \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \delta(x_1) \delta(x_2) f(x_1,x_2) dx_1 dx_2 d \rho$$

where $f(x_1,x_2)$ is the bivariate gaussian distribution. $\rho$ is the correlation coefficient.

$$\int\mathbb{E}[\delta(x_1) \delta(x_2)] d\rho = \int f(0,0) d \rho = \frac{1}{2 \pi \sigma_1 \sigma_2}\int \frac{1}{\sqrt{1 – \rho^2}} \text{exp}\left(-\frac{1}{2 } \frac{1}{1 – \rho^2} \left(v_x^2 + v_y^2 – \frac{2\rho \mu_1 \mu_2}{\sigma_1 \sigma_2}\right)\right) d\rho$$