This Gaussian integral came up while working on a likelihood analysis for pulsar timing arrays:
$$ \int_{-\pi}^{\pi} \exp\left( -(x-y \sin{\gamma} )^2 \right) \mathrm{d}\gamma $$
I've tried everything I can think of, but I can't get an analytic solution.
If you expand the square, the cross term can be written in terms of a cosh, but I don't know where to go from there. That gives the integrand as $\exp{(-x^2)} \exp{(-y^2 \sin^2\gamma)} \cosh{(2 x y \sin{\gamma})}$.
The substitution $\beta\equiv\sin{\gamma}$ leads to $$ \oint_0^0 \frac{\exp\left( -(x-y \beta )^2 \right)}{\sqrt{1-\beta^2}} \mathrm{d}\beta $$
I also tried $\arctan \alpha \equiv \gamma$, which gives
$$ \int \frac{\exp\left[-\left( x- \frac{\alpha y}{\sqrt{1+\alpha^2}} \right)^2\right]}{1+\alpha^2} \mathrm{d}\alpha $$
Neither Mathematica nor Rubi are able to evaluate any of these integrals. I was hoping the residue theorem might be applicable, or one of these substitutions might get into a form that Mathematica knows. Any help will be greatly appreciated and certainly land you in the acknowledgements of the paper my collaborators and I are working on:)
Any approximate results (saddle point approximation?) would also be appreciated.