If you check out the integral tables in section 4.2 and 4.3 of http://nvlpubs.nist.gov/nistpubs/jres/73B/jresv73Bn1p1_A1b.pdf, you will find what you need to get this done. I used equations 4.3.2 and 4.2.1 (latter is same as eqn 7.4.36 in Abranovitz and Stegun). Just change variables on the error function and complete the square on the exponential. You will end up with one 4.3.2 integral, one 4.2.1, and one simple erf(x) integral of the exponential square.
I am a bioinformatician. This problem occured for me in the context of statistics. I was trying to compute conditional probabilities to input in my factor graph model.
I verified equation 4.2.1 from the source litterature. I will have to doublecheck if 4.3.2 is correct (via numerical integration), since this solution is original to this work.