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In this example it helps if you say that your parameters are positive real numbers (are they?)

sage: a,b,c,d,x,k = SR.var('a,b,c,d,x,k')
sage: i = a*(1 + exp(-(b*x)^2/4)*cos(c*x)*cos(d*x))
sage: i0 = i.subs(x=0)
sage: W = 2*i - i0
sage: for v in [a,b,c,d]:
....:     assume(v, 'real')
....:     assume(v > 0)
sage: integrate(W*exp(-I*k*x),(x,-oo,oo))
sqrt(pi)*a*e^(-c^2/b^2 + 2*c*d/b^2 - d^2/b^2 + 2*c*k/b^2 - 2*d*k/b^2 - k^2/b^2)/b + sqrt(pi)*a*e^(-c^2/b^2 + 2*c*d/b^2 - d^2/b^2 - 2*c*k/b^2 + 2*d*k/b^2 - k^2/b^2)/b + sqrt(pi)*a*e^(-c^2/b^2 - 2*c*d/b^2 - d^2/b^2 + 2*c*k/b^2 + 2*d*k/b^2 - k^2/b^2)/b + sqrt(pi)*a*e^(-c^2/b^2 - 2*c*d/b^2 - d^2/b^2 - 2*c*k/b^2 - 2*d*k/b^2 - k^2/b^2)/b