I'd strongly recommend using Strava which is found at http://www.strava.com/
It has options for normal cycling as well as stationary rides on a trainer. An exercise bike isn't physically moving so GPS functions aren't helpful to you, but the static rides will work.
I use an Android S5 with ANT+, and I have a cheap ANT+ cadence sensor for measuring pedal RPMs, and a strap-on ANT+ Heart Rate Monitor which sites around the chest. If your fitness bike is spanky and has a power meter then strava app can read that too, as long as your fancy phone has the correct wireless interface, which is often ANT+ but Bluetooth Low Energy (BLE) is also an option, and is the only option on iphones.
If your fitness bike is a normal bike held in a trainer, then even better because when you remove bike from trainer for a real road ride, the sensors and phone app will be able to log those rides too.
Best thing? Strava's free mode is very nice and contains everything I need. There's a premium mode which offers more bells and whistles for $7 US a month, but that's high end analysis which is beyond an elderly pedal masher like me.
Screenshot from Strava website of recording page (note that one has a HRM, and is showing a running screen) :
Update: Strava removed support for ANT+ devices attached to phones in around 2018, originally for "stability improvement" reasons. Admittedly the app has become more stable over time, but I now have sensors that won't link to the app, and would require some kind of physical head unit on the bike to act as a concatenator/controller.