Let $f(x)$ and $g(x)$ be the PDFs of two Gaussian distributions both with zero mean, and variances $\sigma_1$ and $\sigma_2$ respectively.
I'm trying to compute $$ \int_{-\infty}^{\infty} \frac{f(x) g(x)}{f(x)+g(x)}dx $$ but I am unable to make any progress.
I can also use the fact that the product of two Gaussian densities is an (unnormalized) Gaussian in the same variable to simplify the numerator but this doesn't seem to help.
I would also settle for upper/lower bounds. For example you can bound point-wise $f(x)+g(x)$ with another scaled Gaussian.
We can also assume $\sigma_2 = c \sigma_1$ for some constant $c$ if that's helpful.
Assume $\sigma_1<\sigma_2.$ Writing $$I=\int_{-\infty}^{\infty}\frac{f(x)}{1+\frac{f(x)}{g(x)}}dx=\sum_{n=0}^{\infty}(-1)^n\int_{-\infty}^{\infty}\frac{f^{n+1}(x)}{g^n(x)}dx$$ you land on the series $$\sum_{n=0}^{\infty}(-1)^n \frac{a^nb}{\sqrt{n+c}}$$ where $a,b,c$ are easyly computable, but not the sum of the series.