You don't do it for the profit, at least not at this point. The BitAxe is the cheapest (and most efficient) to run. It's expensive for the hashpower, but the idea is that it'll fund future development of the Open Source Miners United project.
Purely economical, the Canaan packs a lot of punch for the price point. The Braiins miner is a bit in the middle.
BitAxe 0.5 Th / 14W ~$150 (or a lot less if you DIY)
Braiins 1 Th / 40W ~$199 (or €199, unclear at this point)
Canaan 4 Th / 140W ~$150
Most numbers are approximate. I've added them to my calculator, so you can play around with the economics.