What is considered a "complicated computation" can depend quite heavily on the field, and also on the individual mathematician.
But even in fields or papers that don't rely so much on lenghty computations, there might still be lengthy technical arguments, to which the same question applies.
It think it is worthwhile to add the following point to the answers already given.
About checking the correctness of proofs and computations
In my experience many people who start out in mathematical research have a somewhat inaccurate impression of how more experienced people (well, many of them, I guess, though probably not all) tend to check the correctness of mathematical proofs:
Going through every single line of a proof and a computation is not only lenghty and cumbersome, it also tends to be extremely (and by this I mean really extremely) unreliable.
Human minds are not computers and are highly succeptible to various types of mistakes - in particular if we don't produce an argument ourself, but just read an argument that somebody else has written down.
Without a very good intuitive understanding of a proof, it is just too easy to believe that "every step follows from the previous step" - and to overlook a missing assumption in some step, or a wrongly applied theorem, or a plain sign error, or what not.
What is much more reliable (and, luckily, also much more efficient) are certain high-level checks: figure out the crux of an argument, see where a new idea enters the game, test a result against the common counterexamples which you know prevent related statements from being true, see what part of the argument overcomes, in the present situation, the phenomenon that occurs in all these counterexamples, and so on.
I tend to regularly find critical mathematical mistakes when refereeing papers (although I haven't counted the precise percentage where I found such a mistake) - but I don't remember a single instance where I found a mistake by checking every line in a proof, noticing at one step that "this line doesn't follow from the previous line".
In most cases it goes much more like this:
"Ok, Theorem 1 claims this and that. Under the additional assumption A, which is not there, I know that it's true - let me see if I can remember how the argument goes... Ah yes, and for this particular part of the argument I would need assumption A. Now let's see how they do it without this assumption, because this feels like quite a strong claim.
[Start to read the proof, skip the first ten lines because they contain just the usual stuff which one always does to prove results of this type.]
Ah, here's where it get's interesting. They say that claim C is true. Hmm, well, I can see that claim C, once established, will probably imply the theorem. Now let's see how they can get C without A. Ah, ok, here's the crucial argument. Hmm, but I don't understand why this should be true. Let me think again - ok, I have an example which shows that the crucial argument does not work, in general."
This can happen in many variations - but the point is always that, with increasing experience, one often does not need to check every line since one knows how the standard arguments in the field go. So instead one looks for the crucial parts of arguments where things happen that seem to be surprising or non-obvious (to somebody familiar with the standard techniques in the field).
Closely related, and certainly worth a read: Terry Tao's blog post about the three stages of rigour in mathematics.