How to solve this integral analytically or numerically?

Question

I'm trying to evaluate the following integral

$$\int_{-\infty}^\infty \int_{-\infty}^\infty \exp\left(-\frac{(x-x_0)^2}{2}-\frac{(y-y_0)^2}{2}\right)\log\left(e^{-x x_0}+e^{x x_0}+2\cos(x_0(y+y_0))\right) \mathrm dx\mathrm dy$$

where the parameters $x_0$ and $y_0$ are positive real numbers.

p0 = 2

fct = Exp[-(1/2)*(x - x0)^2 - (1/2)*(y - y0)^2] * Log[Exp[-x*x0] + 
  Exp[x*x0] + 2*Cos[x0*(y + y0)]]

intfct = Integrate[fct, {x, -∞, ∞}, {y, -∞, ∞}, Assumptions -> x0 >= 0]

Is there any way to solve the integral analytically or numerically?

Answer 1

Because of the $\log$ term, I doubt that there is any tractable analytic solution. However, numerical integration is immediate:

int[a_, b_][x_, y_] := Exp[-(1/2)(x + a)^2 - (1/2)(y - b)^2] *
  Log[E^(-a x) + E^(a x) + 2 Cos[a (y + b)]]

nint[a_, b_] := NIntegrate[
  int[a, b][x, y], {x, -Infinity, Infinity}, {y, -Infinity, Infinity}]

E.g. try nint[1,2].

One useful transformation for (doubly-)infinite integrals, especially those involving Gaussian functions, is to observe that the integral must be invariant under linear transformation of each variable, e.g. $x\to x-a$ and $y\to y+b$. Applying this transformation to the integrand,

int[a, b][x - a, y + b] // FullSimplify

yields a (somewhat) simplified expression.

One sees that this integral is independent of $b$ for $a=0$ and evaluates to $\pi \log (16)$.

Also, because the transformed integral involves $\cos (a (2 b+y))$—and this is the only place that $b$ appears in the integrand—the integral has to be periodic in $b$ with period $\frac{\pi }{a}$.