Is there a formal proof of non-existence of chain rule formula $C$ for integration?
$$ \int f(g(x))dx = C\big[\int f(x) dx, \int g(x) dx, f(x), g(x) \big] $$
where the question assumes the formula $C$ is finite in length (and independent on any other criteria on behavior of functions $f,g$).
It is clear some functions (e.g. $e^{-1/x^2}$) cannot be integrated within elementary functions. But $C$ might contain some new non-elementary or special functions, so existence of examples of non-elementary integrable functions as counter-example is not a valid argument.
There is a similar question with no answer I would expect:
Integrating composite functions by a general formula?
The non-existence of chain rule is advanced as a basic reason for "hardness" of integration, yet I am not aware of a formal proof.
EDIT: To be more rigorous: In my question $C$ should only contain fuction compostition where some new finite set of (some special) functions can be introduced (and presumabely must be).