Works fine :)
Integrate[7/4 r^3 Cos[17/100 + (4 p)/39]^2 Cos[p/4] Cos[1/10 (33/20 + p)]^2, {r, a, b}, {p, 0, 2 Pi}, {t, 0, 2 Pi}]
(* 2730 (-(a^4/4) + b^4/4) \[Pi] (1/780 + (5 (-1 + Sqrt[5]) Cos[33/100])/
5616 + (195 Cos[1/100] Cos[(2 \[Pi])/195])/304072 + (
195 Cos[67/100] Cos[(37 \[Pi])/195])/494648 + Sin[1/100]/76018 +
1/351 Sin[33/100] - (5 Sqrt[1/2 (5 + Sqrt[5])] Sin[33/100])/
2808 + (8 Sin[17/50])/2485 - (39 Cos[(7 \[Pi])/78] Sin[17/50])/
9940 - (79 Sin[67/100])/123662 - (
195 Sin[1/100] Sin[(2 \[Pi])/195])/304072 + (
39 Cos[17/50] Sin[(7 \[Pi])/78])/9940 + (
195 Sin[67/100] Sin[(37 \[Pi])/195])/494648) *)
Clear the Kernel and then caluculate integral:
ClearAll["Global`*"]; Remove["Global`*"]