Integrating Hard Integrals: No Closed Form Solution

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The integral in question converges under specific parameters, but a closed form solution remains elusive, requiring the use of special functions beyond standard Beta, Gamma, and Error functions. Suggestions include changing the region of integration to simplify the problem, potentially transforming it into a form that can be expressed using Beta or Gamma functions. The discussion invites input on any attempts made to alter the integration region, particularly in the context of a triangular domain within [0,1]x[0,1]. Participants are encouraged to share their findings and methods used in their attempts to solve the integral. The conversation emphasizes the complexity of the integral and the need for innovative approaches to find a solution.
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Please see the attachment. Under the given range of parameters the integral converges, but I can't find a closed form solution. It seems one has to integrate very special functions other than simple Beta, Gamma, Error functions etc..
 

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semigroups said:
Please see the attachment. Under the given range of parameters the integral converges, but I can't find a closed form solution. It seems one has to integrate very special functions other than simple Beta, Gamma, Error functions etc..

Hey semigroups and welcome to the forums.

Have you considered changing the region of integration to get things in terms of one variable?

My guess is if this is a problem from a problem set, that a change of variables and some extra stuff will give you something that ends up being in the form of a Beta (complete or incomplete) or a Gamma (complete or incomplete) and then you can just leave it at that.

If you have tried changing the region (I think it should look something like a triangle in the [0,1]x[0,1] region) then what did you try exactly and what did you find out?
 
Thanks for reply!
Please find the attachements for deatailed working (due to the size of screen capture I have to divide it into three pictures).
 

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