Question: inverse fourier transform

I have a system of linear differential equations and am trying to solve them using Fourier transforms.

I can reduce the system to a final result (for the variable of interest) in Fourier space as this (note the frequency variable is 'w'):

restart:

with(inttrans):

vout_fourier_num := fourier(phi[3](t), t, w) = 6.63569999999998*10^(-15)*w^2*fourier(V(t), t, w)/(-5.69875218358308*10^(-40)*w^4+(9.19473390627057*10^(-29)*I)*w^3+2.15219369729956*10^(-18)*w^2-(4.14691648617110*10^(-8)*I)*w-700.8);

#the drive can be defined as:
drive:=5.70000000000000*10^(-6)*exp(-3.18877551020408*10^18*(t-2.0*10^(-9))^2)*cos(4.8*10^9*Pi*(t-2.0*10^(-9)))*10^(19/20);

#substitute the drive in - this is not necessary, but it should work!...
vout_fourier_num2:=subs(V(t)=drive, vout_fourier_num);

#now take the inverse... note this gives 0!...
invfourier(vout_fourier_num2, w, t);

the final results calculated is zero. It is wrong... it seems like an accuracy issue, but increasing the digits does not help. I should note that I can calculate the solution directly via dsolve, and get completely reasonable answer.

any ideas how to get the Fourier method to work?

thanks!