Even for integrals, derivation isn't a silver bullet. You need to derive and also test for equality. Determining whether two expressions are the same (or enough the same, like a being the same as a^2/a except for a=0) can be really hard.
Can't you just check numerical equality at a few million random points?
is it usually the case that derivatives are easier to compute than integrals?
The general algorithm for calculating integrals [1] is rather complex and I guess not suitable for humans so that calculating integrals sometimes looks much more like a black art then calculating derivatives does. On the other hand one could argue that there are algorithms for doing both and so there is no real difference.
Yes. Most of the time, the reason is that there is a general form for computing derivatives of products ((xy)' = x'y + xy') and composite functions (chain rule) while no such general rules exist for integrals.
Symbolically, yes that's almost always the case. Numerically it's the other way around.