What is the probability $P(X>0\mid X+Y>0)$ when $X$ and $Y$ are both independent normal mean zero with different variances?

Question

Let $X$ have standard deviation $\sigma$ and $Y$ standard deviation $\tau$. I can write $$P(X>0\mid X+Y>0) = \frac{P(X>0, X+Y>0)}{P(X+Y>0)}$$ As all variables are zero-mean and symmetric, $P(X+Y>0)=1/2$, so that $$P(X>0\mid X+Y>0) = 2 P(X>0, X+Y>0) = 2 P(X>0, Y>-X)$$ Using the independence of $X$ and $Y$ and the notation $\Phi$ and $\phi$ for the standard normals, I get $$P(X>0\mid X+Y>0) = 2 \int_0^\infty \phi\left(\frac{x}{\sigma}\right) \left(1-\Phi\left(-\frac{x}{\tau}\right)\right) dx.$$ But from here on I do not know how to continue to get a simple expression. Any suggestions?