First of all, I have taken a look at the Grinter's method, which is similar to Cross, but balancing angles instead of moments. My question assumes that we are using the Cross method, and not the Grinter's method.
The literature explaining the moment distribution (i.e: Cross) method for statically indeterminate structures tends to affirm that the designer needs the moments rather than the rotation angles, and therefore all books and references I checked don't pay any attention at all at getting the rotation angles. I think that's a false affirmation, because the designer needs to check both ULS and SLS requirements (with SLS being quite often more limiting than ULS), and you need to know the rotation angles in order to find the maximum deflection and be able to check it.
It's a bit surprising, because the "we just need the moments" affirmation seems to suggest that engineers didn't check the maximum deflection in structures designed with the Cross method, which allegedly was broadly used from the 1930s to the 1960s. I find it hard to believe that they didn't check the maximum deflection in their structures.
I'm wondering what would be the most efficient way of finding the joints rotation angles after you apply the Cross method. Do you really need to apply the Grinter's method instead of Cross? Is it not possible to get the rotations starting from just the Cross solution?
The only solution that came to my mind is to integrate the moments diagram twice, but the conditions for solving the integration constants would depend on the structure topology, and I wouldn't call it "efficient"...