Why there's no chain rule for integrals of elementary functions which are expressible in terms of elementary functions? The derivative of every elementary function is elementary; this is owing to the existence of the chain rule for differentiation. 
On the other hand, the integral of an elementary function may turn out to be elementary or not elementary ($\text{e.g:}\int e^{-x^2}dx$). There's Risch algorithm, which for a given  integral of an elementary function, tells you whether the integral is elementary or not, and if it's elementary, it finds the solution.
However I think it's still valid to ask, for integrals of elementary functions that are expressible in terms of elementary functions, why there's no chain rule for them? 

Differentiation is a function that satisfies linearity f(x + y) = f(x) + f(y) and f(ax) = af(x). It also satisfies the rule f(xy) = f(x)y + xf(y). Integration can be thought of as the inverse function much like division can be thought of as the inverse function to multiplication. However, just as division has only some of the same algebraic properties as multiplication but not all, it is not commutative, nor is it closed on a set with zero, integration does not have all the algebraic properties of differentiation. Indeed the reason for this is precisely because of division not having all the algebraic properties as multiplication. I say "precisely", making this a rigorous logical proof is I'm afraid beyond me.
