Taylor series of integral of exponential of Fourier series

I have experimented with integrals of

exp(i·f(t))

where ”f(t)” is a polynomial. I express these integrals as Taylor series, which are not hard to generate. Now I’d like to try

f(t) = a·t + Σb_k·sin(k·t)

and I have a horrid feeling that the Taylor series will be a nightmare!