I would like to calculate the following integral
$$ \displaystyle\int_0^{T} e^{-s\left[t+\frac{1}{\mu}(e^{-\mu r }- e^{-\mu(r-t)})\right]} dt$$
where $s,\mu, r$ are constant values.
What I have done:
after some algebra I got that this integral is equivalent to
$$ c\displaystyle\int_0^{T} e^{-st} e^{-c e^{\mu t}}dt $$
where $c = e^{-\frac{s}{\mu} e^{-\mu r}} $.
Now, from my understanding this integral is not possible to solve it analytically, but I am not 100% sure since I do not know how to prove that.
Any insights are appreciated.