Appeard in SIAM J. Math. Anal. 25 (1994) [Askey issue], 812-814.
The celebrated WZ method can prove, fully automatically any multi-variate `nice' integral evaluation, for any FIXED dimension. A very nice implementation was written by Akalu Tefera (The Maple package Mint). With a tiny huamn help, it may be able to prove it for GENERAL dimension.
Doron Zeilberger's List of Papers