I get the author's point but this is not completely true; there exist functions that are not differentiable at certain places (e.g. ideal square waves) and others that are not differentiable anywhere (e.g. Weierstrass functions).
In general, to even ask what it means to compute a derivative we need to specify some input language which describes functions in finite terms; we are necessarily in the world of constructions rather than (say) arbitrary set-theoretical maps between infinite sets. With this in mind, the claim that differentiation is always a straightforward computation is a strong one.
With integration, there's often no closed-form process at all, as the author points out.
Reminds me of a homework exercise in theory of computation back in my uni days. Is the function "f(x) is 0 if H else 1" computable? Of course it is. It is either constant 0 or constant 1. We don't know which at this point, but f is a constant function in either case, so it is surely computable.