I mean, sure, we have simpler rules for dfferentiation, but _why_?
I sometimes wonder if it's differentiation is P and integration is NP (for the restricted case of functions where the primitives are "nice")
The most straightforward definition of indefinite integration on the other hand is “the inverse of differentiation”. Like a lot of functions, inverses are much more complicated. Even in the case of matrix-vector multiplication, the inverse function (the inverse matrix) has an expression in terms of determinants in the original entries of the matrix - much more complicated than the original problem.
So I think that it’s the more simple theme of: even if the function is easy to describe, its inverse may be extremely complicated.
The more abstract reason differentiation is easier than integration is that because differentiation is a local operation, given a family of functions differentiation will be closed within that family. That is the derivative of an elementary function is an elementary function, the derivative of a rational function is a rational function, the derivative of a function f is a function f' that is within the same family of f and will share almost all of the same properties as f.
Integration on the other hand does not have this property, and in fact it's very difficult to find family of functions that are closed under integration. The integral of a rational function may not be rational, the integral of an elementary function may not be elementary. Integration is a global operation whose solution may require the construction of a new family of functions whose properties are entirely unlike the function being integrated.
It's like how squaring a number is usually closed within that number system, but taking the square root of a number could require creating entirely new number systems, whether it's real numbers or complex numbers (at which point the square root is closed). With integration, the issue is that it never ends up being closed, you can use integration on a family of functions to jump out of that family and construct entirely new families of functions seemingly without end.
In a way, differentiation produces functions whose properties are a subset of the original function, but integration produces functions whose properties expand upon the original function.
It's something pretty close to this. Almost all mathematical expressions can't be integrated (up to the reader to formalise the definitions here). That is not quite the same claim as saying that the best method for solving an arbitrary integral is trial and error by systematically differentiating other expressions until you find the antiderivative.