One issue is that for differentiation, you can find the derivatives of so-called "elementary functions" (e.g. powers, exponentials, logarithms, trig functions) directly from the definition of a limit, and there are then standard ways to find the derivatives of an expression knowing the derivatives of its component parts (e.g. the product and quotient rules, function-of-a-function, etc.)
Therefore it is straightforward to generate a large number of "textbook exercises" of graded difficulty.
On the other hand, most "simple-looking functions" do not have closed-form integrals, and the easiest way to find a set of basic functions that do have closed form integrals is to recognize that "the function you want to integrate looks like a derivative that you already know".
In fact many of the so-called "special functions" in mathmematics are *defined" as the integral of a simple-looking function (e.g. the gamma function, which generalizes the idea of a factorial for non-integer values).
Of course as other answers have said, if you taught integration using only numerical methods as an initial way to evaluate integrals, that problem does not exist, but (at high school or university level) most students will not have much if any understanding of numerical methods, and therefore there are too many possibilities for "garbage-in garbage out" exercises where students have no way to check their work.
Typing formulas into Wolfram's integral evaluator isn't "learning math" - and it isn't even an interesting task, unless you are going to do something interesting with the output.