Substitution reverses the chain rule and integration by parts reverses the product rule. Why does integration have a technique per differentiation rule rather than one general method?

B. Because differentiation is mechanical and integration is pattern matching.

Differentiating is algorithmic — identify the outermost structure, apply the rule, recurse — with never a choice, which is why symbolic differentiation is easy to program. Integration runs it backwards and requires recognising which rule could have produced this expression. The deeper fact is that many elementary functions have no elementary antiderivative.