The integral is extremely difficult to compute, but the analytic result for the mean tetrahedron volume is
(3)
(OEIS A093524; Zinani 2003). Note that the result quoted in the reply to Seidov (2000) actually refers to the average volume
for tetrahedron tetrahedron picking.
Do, K.-A. and Solomon, H. "A Simulation Study of Sylvester's Problem in Three Dimensions." J. Appl. Prob.23, 509-513, 1986.Seidov,
Z. F. "Letters: Random Triangle." Mathematica J.7,
414, 2000.Sloane, N. J. A. Sequence A093524
in "The On-Line Encyclopedia of Integer Sequences."Zinani,
A. "The Expected Volume of a Tetrahedron Whose Vertices are Chosen at Random
in the Interior of a Cube." Monatshefte Math.139, 341-348, 2003.