Consider the following integral
z = 3*Exp[I*\[Theta]]; n = 20; k = 1.25 \[Pi]; p = 1.5; Integrate[BesselJ[(2 n + 1)*p, k*Abs[z]], {\[Theta], -\[Pi]/2, \[Pi]/2}] However, when I replace Abs[z] with 3, it gives the correct answer:
Integrate[BesselJ[(2 n + 1)*p, k*3], {\[Theta], -\[Pi]/2, \[Pi]/2}] 
$Version (* "13.2.1 for Mac OS X ARM (64-bit) (January 27, 2023)" *) Clear["Global`*"] It is a precision issue . Use exact values for the constants and use arbitrary-precision rather than machine precision.
z = 3*Exp[I*θ]; n = 20; k = 5/4 π; p = 3/2; Block[{$MaxExtraPrecision = 100}, Integrate[BesselJ[(2 n + 1)*p, k*Abs[z]], {θ, -π/2, π/2}] // N[#, 15] &] (* 1.04339529077324*10^-37 *) 