Thanks a million. I have tried the way you said, however, I want to get a general parameter solution.

Best wishes to you all!

The expression is very complicated for Mathematica.

For general parameters, Mathematica is struggling to calculate the integral.

If we assume:

Subscript[m, e] = 1; Subscript[p, w] = 2;

Assuming[{L > 0, Subscript[a, e] > 0, I0 > 0, Subscript[c, e] > 0, 
  Subscript[c, w] > 0, Subscript[\[Alpha], e0] > 0, 
  Subscript[\[Alpha], w0] > 0}, 
 FourierCosCoefficient[(I0 + Subscript[a, e]*Cos[\[Theta]])*(1 - (
     Subscript[\[Alpha], e0]*Subscript[c, e]*
      L)/((Subscript[m, e]*Cos[\[Theta]])^2 + 1) - (
     Subscript[\[Alpha], w0]*Subscript[c, w]*
      L)/((Subscript[p, w] + Subscript[m, e]*Cos[\[Theta]])^2 + 
      1)), \[Theta], 2]]

 (*-(1/(20 Sqrt[2] \[Pi]))
  L (40 (-3 + 2 Sqrt[2]) I0 \[Pi] Subscript[c, e] Subscript[\[Alpha], 
      e0] + (I0 (((-42 + 4 I) Sqrt[-1 - 2 I] + 
              80 Sqrt[2] - (6 + 26 I) Sqrt[20 - 10 I]) \[Pi] + (2 + 
              21 I) Sqrt[-1 - 
             2 I] (-2 I ArcTan[1/2] + Log[5] + 
              4 Log[-Sqrt[-(2/5) - I/5]])) + (((88 + 34 I) Sqrt[-1 - 
                2 I] - 320 Sqrt[2] + (38 + 46 I) Sqrt[
               20 - 10 I]) \[Pi] - (44 + 17 I) Sqrt[-1 - 
             2 I] (2 ArcTan[1/2] + 
              I (Log[5] + 4 Log[-Sqrt[-(2/5) - I/5]]))) Subscript[a, 
         e]) Subscript[c, w] Subscript[\[Alpha], w0])*)