# Trying to Integrate this Non-Integrable Function

Are there potential methods to solve this non-elementary integration problem? At the beginning of break my Calc BC teacher has given us this problem to use "all necessary resources" to see if we can indeed figure out a way to, even theoretically, crack it:

$$\int e^{\sin x}\ dx\ =\ ?$$

I've worked towards transforming it a little using some differential equations, such as the following: $$y = e^{\sin x},\quad y'=\cos x\left(e^{\sin x}\right)\\ \int y'\sec x\ dx = \int y\ dx \\ y\sec x\ -\ \int y'\sec x\tan x\ dx = \int y\ dx$$ However, I'm having trouble getting anywhere with it. Just looking at the graph and its (obviously) cyclical qualities begs me to find the function that defines its antiderivative. It just looks like it has one! Surely, of course, I am mistaken, but I want to have it proven to me that, having exhausted all of the possibilities, there is no antiderivative.

Please help me in uncovering the mystery of this integral!(that is, if the possibility of surprising my teacher does not entertain you...) Of course, I appologize for the naivety of my endeavor, but I believe it is my duty as a learner of calculus to be exploratory if not scrutinous of the many claims my teacher makes to prove to myself that it is in fact true.