I have seen two ways of saying "I can't complain":
Ich kann nicht klagen.
Ich kann mich nicht beklagen.
Do they mean the same?
If so, which one is more commonly used?
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Yes, the meaning is identical. The first sentence could have a different meaning (lawsuit), though.
Which is more common is hard to answer. My personal opinion is that "Ich kann mich nicht beklagen" is more common.
According to Ngram Viewer, however, their uses are almost equally frequent; with a slight tendency towards "Ich kann nicht klagen".
You're fine either way.