actually, one gets a very simple solution in the following way:

1) assume w > 0.

2) rewrite the cos as the sum of two complex exponentials

3) each of the two integrals involving the exponentials has the structure of a Fourier transform of exp(-(x-x0)/w^2)

and use a simple coordinate transformation x->x+x0 plus the easily found fact

int(exp(I*p*x)*exp(-x^2/w^2),x=-infinity..infinity)=

exp(-1/4*w^2*p^2)*|w|*Pi^(1/2)

that basically says that the Fourier transform of a Gaussian is a Gaussian of essentially reciprocal width.

I obtain the result w*Pi^(1/2)*exp(-k^2*w^2)*cos(2*k*x0) after simplification.

So, a little thinking may help, and there is no need at all to use symbolic algebra here.

But what about non-real w? The answer is to use analytic continuation in w here from the result that is valid for w>0 to a larger region which is such that the original integral converges. This would require that Re(1/w^2)>0.

Probably, one may regard the integral as distribution in the variable k. Then, it seems to make sense for all nonzero w.

Setting w=1

Integrate[ Cos[2 k x] Exp[-(x - x0)^2/w^2], {x, -\[Infinity], \[Infinity]}]

The answer is:

1/2 Sqrt[\pi]] (1 + e^(4 i k x0)) e^(-k (k + 2 i x0))

Thanks to you all!

The general answer is :

Sum_{n=0}^{infinity} [ {2k^{2n}/(2n)!!} Sqrt[\pi] (omega )^{2n+1} ]

actually, one gets a very simple solution in the following way:

1) assume w > 0.

2) rewrite the cos as the sum of two complex exponentials

3) each of the two integrals involving the exponentials has the structure of a Fourier transform of exp(-(x-x0)/w^2)

and use a simple coordinate transformation x->x+x0 plus the easily found fact

int(exp(I*p*x)*exp(-x^2/w^2),x=-infinity..infinity)=

exp(-1/4*w^2*p^2)*|w|*Pi^(1/2)

that basically says that the Fourier transform of a Gaussian is a Gaussian of essentially reciprocal width.

I obtain the result w*Pi^(1/2)*exp(-k^2*w^2)*cos(2*k*x0) after simplification.

So, a little thinking may help, and there is no need at all to use symbolic algebra here.

But what about non-real w? The answer is to use analytic continuation in w here from the result that is valid for w>0 to a larger region which is such that the original integral converges. This would require that Re(1/w^2)>0.

Probably, one may regard the integral as distribution in the variable k. Then, it seems to make sense for all nonzero w.