Which software is appropriate to integrate radial velocity with respect to time
I'm not sure what you are actually trying to do. Radial velocity isn't generally measured, it is computed from the projected velocity. But if I take you at your word, that your raw data is radial velocity, then that simplifies things. 

If you know the period, $P$, and you know the $v$ AND if you know the theoretical relationship between them $v = f(p)$ then you can use linear least squares to compute the adjustable parameters of $f(p)$ by plotting $v$ (measured) vs $v$(calculated). 

Any 'best fit' equation is able to interpolate values of $v$ so all you need to divide up the period into small intervals and approximate the area under the curve using its two end-points (it is rarely necessary to use parabolic fits that also use the mid-points of those small intervals). 

Thus the problem of integration becomes one of simple geometry (you have a tall thin rectangle with a triangle on top of it for each interval). Let's think about a star with a 30 day period. That's ~40,000 minutes, so I'd probably use 5 or 8 minute intervals ...starting at minute 0 what's the area under the curve from t=0 to t=8 minutes? then from 8 to 16, all the way up to 43,200th minute. Sum of those areas is total area. An Excel spreadsheet can easily do that. 

Actually, if you also use 5 minute intervals and compare the results you'll see how much better (hopefully NOT much) a smaller interval is...analytical integration is this taken to the limit of the interval approaching zero, but in real life, numerical integration is necessary for almost all 'interesting' problems.  Here are 2 links, fwiw: 


https://www.physicsforums.com/threads/finding-periodic-best-fit-equation-for-data-set.715638/
http://www.aanda.org/articles/aa/pdf/2004/08/aah4718.pdf ← see page 537 and references noted there.
The Institute for Astronomy (IFA) at the University of Hawaii has several pdfs on BW that may be helpful