I think that the referenced paper has all you might want (it is obviously related to the Derevnyn-Mednykh paper, but is considerably more extensive).They do not specifically talk about orthoschemes, but their results specialize nicely.

@article {MR2265285,
    AUTHOR = {Mednykh, A. D. and Pashkevich, M. G.},
     TITLE = {Elementary formulas for a hyperbolic tetrahedron},
   JOURNAL = {Sibirsk. Mat. Zh.},
  FJOURNAL = {Rossi\u\i skaya Akademiya Nauk. Sibirskoe Otdelenie. Institut
              Matematiki im. S. L. Soboleva. Sibirski\u\i\
              Matematicheski\u\i\ Zhurnal},
    VOLUME = {47},
      YEAR = {2006},
    NUMBER = {4},
     PAGES = {831--841},
      ISSN = {0037-4474},
   MRCLASS = {51M09},
  MRNUMBER = {2265285 (2007e:51023)},
       DOI = {10.1007/s11202-006-0079-5},
       URL = {http://dx.doi.org/10.1007/s11202-006-0079-5},
}