I found this Hotjar service very interesting. It has a free plan, and these features:
- Conversion funnels
- Form analysis
- Feedback polls
- Recruit user testers
Even if you won’t use it I find this article, Action Plan, very useful to understand what brings traffic to your app, and what the barriers to make people in or drive people out.
Hotjar has a integration with Optimizely, and as I know no API.