I guess you can find plenty of them in the Problem Department of the American Matematical Monthly. For instance, here is a nice one by M. Hajja and P. Walker:
Evaluate $\displaystyle \int_{0}^{1}\int_{0}^{1} \int_{0}^{1} (1+u^{2}+v^{2}+w^{2})^{-2} du dv dw $.
Good thing about this one is that there may exist a CAS out there that does not give the answer at once.