Has it been proven that there is no formula for analytic integration? Is this still open, could there be a formula?
I assume the question is "differentiation has formulas that you can just plug functions into to get your answer. Do these exist for integration?"
In which case the answer is no, though as previously stated the Risch algorithm gets you most of the cases where they do exist. Try solving $\int \sin(x^2)$, for example (but don't try too hard).