Reverse in the sense that just like chain and product rules let us find the derivative of f(g(x)) and f(x)×g(x) respectively, something that lets us find the anti-derivative of a function composition or the product of two functions.
There isn't any specific formula for reversing the chain or product rule, but it can usually be accomplished by using u substitution in an integral (anti-derivative).
Not in that easy way like with differentiation. For example partial integration uses the product rule to manipulate the integral of a product to hopefully get something easier, substitution can be used with some composite functions.

In general finding the anti derivative is a lot harder than finding the derivative and often it's impossible to find a closed form.
Integration by substitution is the procedure which in essence reverses the chain rule.

Integration by parts is the procedure which in essence reverses the derivative of product rule.

However if you're looking for rules which would give you the antiderivative of composition/product, then tough luck — no neat formula there.