Is this Integrand of a Complex Integral Even or Odd?

[quadrant]

I'm trying to do an integral of the form

$\int_{-\infty}^{\infty} e^{-Ax^2} e^{-iBx} e^{iCx^2e^{-Ax^2}} dx$

I've tried differentiating the integrand to see if I could get something useful but didn't get anywhere. I've been trying to do it for ages and it's driving me crazy! If someone could give me a hint or a prod in the right direction that would be really helpful.

 

[mateomy]

This might be a stab in the dark, but have you checked to see if the function itself was even or odd?