calculus and analysis – How to prove the following integration identity?

I have the following integration that I want to evaluate it using Mathematica.

Evaluate(Sqrt((Theta))/(Sqrt(p ) BesselK(1, Sqrt(p (Theta))))
   Integrate(
   x BesselK(0, 
     x Sqrt(p (Theta))) (Sqrt((Theta))/Sqrt(
      s ) (rTD BesselK(1, 
          rTD Sqrt(s (Theta))) (BesselI(0, x Sqrt(s (Theta))) - 
           BesselK(0, x Sqrt(s (Theta))) BesselI(0, Sqrt(
             s (Theta)))/BesselK(0, Sqrt(s (Theta))))
        + 
        BesselI(0, 
          x Sqrt(s (Theta))) (x BesselK(1, x Sqrt(s (Theta))) - 
           rTD BesselK(1, rTD Sqrt(s (Theta)))) HeavisideTheta(
          x - rTD)
        + 
        BesselK(0, 
          x Sqrt(s (Theta))) (x BesselI(1, x Sqrt(s (Theta))) - 
           rTD BesselI(1, rTD Sqrt(s (Theta)))) HeavisideTheta(
          x - rTD))), {x, 1, Infinity}))

I find problem in obtaining the solution using Mathematica, so I tried to evaluate it manually.

I want to make sure that my solution is correct. My solution is as follows.