The only way an RSX 6spd transmission is doing to fit in your DA Integra is if you bring the K-series motor along for the ride.

Quaife may make a 6 spd gearset for the S80 transmission, but I'd expect it to cost $3500 or more.

Live with the transmission you've got.
A 6spd car will not necessarily be faster than a 5spd.

If you've got $8K to burn and want to swap in a K20A2, go on over to the Hybrid forum.