The stuff coming out of those computer algebra things is not always exactly the form you want to be stuffing in your code. Working it out manually focuses the mind.

They also tend to do it quite badly in some non-trivial cases. I've seen Wolfram spewing a three-line result which could be further simplified to a simple expression. It also doesn't always help to simply know the result; a lot of insight about numerical problems can be gained from how the result is derived.

I also spent several days churning formulae in bars. Got a lot of weird looks.

Don't worry, you have them. :)