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}, }

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.

A. D. Mednykh and M. G. Pashkevich, "Elementary formulas for a hyperbolic tetrahedron", Sib. Math. J. 47 (2006), 687-695.