I believe you're using the wrong residue for the pole at based on the branch you're using: You define:

Then for

Also, try it first for and or something easy like this. Do it numerically first in Mathematica, then do it numerically via the Residue Theorem:

Code:

In[56]:=
p = 1/4;
t = Pi/4;
NIntegrate[1/(x^p*(1 - 2*x*Cos[t] +
x^2)), {x, 0, Infinity}]
r1 = Exp[-7*Pi*(I/16)]/
(Exp[(-Pi)*(I/4)] - Exp[Pi*(I/4)]);
r2 = Exp[(-Pi)*(I/16)]/(Exp[Pi*(I/4)] -
Exp[(-Pi)*(I/4)]);
N[((2*Pi*I)/(1 - Exp[-2*Pi*I*p]))*
(r1 + r2)]
Out[58]=
3.4907507252185646
Out[61]=
3.4907507252151033 +
3.487868498008632*^-16*I