Just to add another bit of colour into the thread - we (AeroCoach) make noncircular 1x and 2x chainrings for TT and road called ARC rings, and I designed the shapes based on a whole heap of pedal mechanics data during my academic career.
The things to note with the ARC rings are that the shape is very subtle, it changes depending on the size of the chainring, and also the shape is not an oval. I'll run through why that's the case along with some general notes:
1. First thing to note is that you're not going to get a 10w improvement (or even close to it) with a non round chainring. If you see that from the get go it is quite likely to be a power inflation from a very noncircular ring (the more noncircular the more the chance of power inflation with a power meter).
2. Secondly, a shape that is very different from round is going to take some adaptation, which then over time affects your pedal stroke as you become accustomed to it. My recommendation if you like noncircular rings that have quite a dramatic shape is just to use them for racing but don't train on them.
3. Our rings change shape only very slightly, which has the side benefit of allowing you to race and train on them and swap between ARC and round without an adaptation period.
4. Crucially, we change the shape of the ring based on the inertia you're likely to experience because your pedalling mechanics change with inertia - at high inertia (big chainrings/high speed) the maximum effective gearing is only 0.5t larger than the average as your pedal stroke is more consistent (=less "choppy"). So a 60t ring goes up to 60.5t max (and 59t at minimum). With our smaller rings the difference is greater, as you're more likely to be riding at slower speed with a 40t or 44t inner where your pedal speed changes more during one revolution. You'll notice going uphill at slower speeds how it's a more choppy pedal stroke, perhaps with the occasional clunk as you lose engagement in the freehub whilst moving through the top of the pedal stroke, that kind of thing.
5. Because everyone is different, and because during the pedal stroke you hold peak force for a while rather than hitting it and immediately decreasing, we also hold the peak gearing period on the chainring over a longer phase than an oval ring (which hits peak and then immediately drops off the gearing), so that we ensure we can "capture" that peak regardless of your pedalling mechanics, as well as maintaining that peak gearing for as long as you're applying peak force.
6. On an ARC ring there are 8 different changes in gearing within one revolution, with a steep increase in gearing in advance of the power phase to mimic the sharp increase in muscular activity of the vastus lateralis, medialis and to a lesser extent tibialis anterior. After the peak gearing phase there is a smoother dropoff as the gastrocnemius and soleus (and again for some people tibialis anterior) come into play and bring your leg through the lower dead spot.
7. Anecdotally our customer feedback is that the feel of an ARC ring is a bit like a flywheel once you get it up to speed, but again it is quite subtle (and you want it to be subtle).
So really it's not a magic bullet, but certainly something that customers like for for sustained aerobic efforts as the ARC ring gearing is matched to how your muscles are firing. They're also made from carbon which results in a bit lower friction and looks awesome.
AeroCoach UK
http://www.aero-coach.co.uk