See here. WolframAlpha should have given ln|x|, but it didn't. Your answer is better than WolframAlpha's, I would say.
I recently did an integration that came up with an answer different from what WolframAlpha comes up with. Here's the integral and my answer:
Here's what WolframAlpha comes up with. (And here's the link.)
The expressions are not the same. I'm inclined to believe WolframAlpha over my own work, but I can't see an error in my work, either. It's very basic, but I'll provide, for example, the case when x is negative:
The integration for positive x is similar.
Did I do anything wrong?
-Dan