One way to think of this is to realize how easy it is to multiply numbers but how much more work it takes to divide numbers.
For something like automatic differentiation, you're essentially applying the chain rule for partial derivatives repeatedly. This is analytically pretty straightforward to do for most applications. All you need is an analytical derivative for the simple functions your more-complex function is comprised of (e.g. a neural network).
For integration, the analogue of the chain rule is integration by substitution [1]. The toolbox for solving integration problems is more limited than for differentiation. You run into issues where the answer cannot even be expressed using standard mathematical notation [2]. Sometimes you get lucky and the answer can be expressed via an alternating Taylor series so you can estimate the answer within some margin of error [3].
Stan is a piece of software that runs state-of-the-art MCMC methods to basically just compute fancy integrals. A Stan model will take an order of magnitude more time to run than a simple neural network via something like PyTorch on the same dataset. But they answer different questions.
[1] https://math.stackexchange.com/questions/1635949/is-there-a-...
[2] https://math.stackexchange.com/questions/1397132/why-cant-so...
[3] https://math.stackexchange.com/questions/145087/how-to-calcu...