Tool for measuring Real and Reactive Power
There are power analyzers specifically meant to do exactly what you are looking for.

One product is the Fluke 435 Power Logger: FLUKE 435

Unfortunately these types of products are not "cheap".  It's not a trivial measurement to make, since it's not just a matter of volts/amps/phase angle but harmonic distortion as well.

Another method as you suggest is to log the current and voltage at the same time, and post-process with Matlab.  Then you could just use any one of a bunch of PC data acquisition modules and a current transducer.  This is probably the cheapest solution.  However, your sample rate would have to be several times the highest harmonic that you want to take into account, so maybe a couple of kHz.  You could then have Matlab spit out the real and apparent power at 1 sec intervals.