I want to integrate a trigonometric function where the argument contains an exponential function. $$f(t) = \cos\left(\omega_{0} \left(1 - e^{-t/\tau} \right) t \right)$$ where $\omega_{0}$, $\tau$, and $t$ are all real and positive. The end game is to use this in Fourier Transforms.
Does anyone know any integral identities, or integration tricks to perform this? I have tried using a series expansion of $e^{x} =\sum1 + x^{n}/n! $ to a fixed number of $n$ to approximate but this doesn't help either.
So what in the end I want to achieve is the following, $$\int_{0}^{T} \cos\left(\omega_{0} \left(1 - e^{-t/\tau} \right) t \right) \ dt $$ and then finally if I can, $$\int_{0}^{\infty} \cos\left(\omega_{0} \left(1 - e^{-t/\tau} \right) t \right) e^{i \omega t}\ dt $$
One of the main problems we have to face is the quite unusual argument of the cosine function. For the sake of simplicity we will omit the role of the two occuring constants by letting $\omega_0=\tau=1$ $($note, the crucial point of what follows is not affected by this decision$)$. Now apply the difference formula of the cosine function to get
$$\cos\left(\omega_0t(1-e^{-t/\tau})\right)\stackrel{\omega_0=\tau=1}\mapsto\cos\left(t(1-e^{-t})\right)=\cos(t)\cos(te^{-t})+\sin(t)\sin(te^{-t})$$
We will now focus on the first summand hence a similiar strategy can be exploited for the second summand too. Recall Euler's Formula to rewrite the cosine in terms of complex exponentials. We are left with terms of the form
$$\large e^{it(1+e^{-t})},~e^{it(1-e^{-t})},~e^{-it(1+e^{-t})},~e^{-it(1-e^{-t})}$$
Essentially these different exponentials are of the form "$x^{-x}$" $($apply the substitution $e^{-t}\mapsto t$ to see what I am refering to$)$. This function, however, is an example of functions for those we cannot find an anti-derivative in terms of elementary functions. Eventually this is the reason why it is hardly possible to find a closed-form anti-derivative of your given function. Nevertheless, it might be of help to actually perform the aforementioned substitution and expand the exponential as a series afterwards which would be similiar to the proof of the so-called Sophomores's Dream in order to obtain a reasonable approximation.
Concerning your end game of finding the Fourier Transform of this function: adding the factor of $e^{i\omega t}$ does not alter the complicated inherent structure of the integrand from which it might be concluded that there is no expression in terms of known functions aswell $($in fact, WolframAlpha fails again to find a suitable expression$)$. Even though it might be easier to exploit numerical methods here since we are dealing with a definite integral rather than an indefinite.