Fateman: Log[Tan[x/2 + Pi/4]] - ArcSinh[Tan[x]]

Differention (and using Maple) can give it as zero, though that is not a blackbox 'simplification command'

http://en.wikipedia.org/wiki/The_Guide_for_the_Perplexed

Thx, the word choice in the article is funny for a paper :-)