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I am reading a publication and trying to follow all of the derivations, but I am having trouble evaluating the following integral which they say has a solution!

Integrate[(1/2 (phiM + x + y)^2 + 
phiM^3/24 (3 phiM^2 + 12 phiM*x + 12 x^2)) (((phiM + x + y)^2 + 
 phiM^2 (3/4 phiM^2 + x^2 + 2 x*phiM + y*phiM + 2 x*y))/((phiM + 
    x + y)^2 + (1/4 phiM^2) (phiM + 2 x)^2)^3 - 
1/((phiM + x + 
    y)^2 + (1/12 phiM^2) (phiM^2 + 4 x*phiM + 4 y*phiM + 
     12 x*y))^2), {y, 0, Infinity}]

Sorry I know it looks like a mess, but the integral is integrated over y from 0 to Infinity and x, phiM are both constants of integration. Any help would be greatly appreciated!

Below is the integral in a more simpler view:

enter image description here

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  • $\begingroup$ Adding the equation in the math form might help. $\endgroup$ – anderstood Nov 3 '16 at 20:27
  • $\begingroup$ @anderstood ok done! $\endgroup$ – user1886681 Nov 3 '16 at 20:40
  • $\begingroup$ Any assumptions on x, phiM? Real? Positive? $\endgroup$ – Michael E2 Nov 3 '16 at 21:06
  • $\begingroup$ @MichaelE2 I've tried those assumptions in the integration but it did not help :( $\endgroup$ – user1886681 Nov 3 '16 at 21:13
  • $\begingroup$ Who is this they? $\endgroup$ – Feyre Nov 3 '16 at 21:36
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I successfully integrated it with the assumptions as follows:

    int=Integrate[(1/2 (phiM + x + y)^2 + 
    phiM^3/24 (3 phiM^2 + 12 phiM*x + 12 x^2)) (((phiM + x + y)^2 + 
       phiM^2 (3/4 phiM^2 + x^2 + 2 x*phiM + y*phiM + 
          2 x*y))/((phiM + x + 
           y)^2 + (1/4 phiM^2) (phiM + 2 x)^2)^3 - 
    1/((phiM + x + y)^2 + (1/12 phiM^2) (phiM^2 + 4 x*phiM + 
           4 y*phiM + 12 x*y))^2), {y, 0, Infinity}, 
 Assumptions -> {phiM > 0, x > 0}]

using Mma11, Win7. The result is here but it is huge. Here is an image of a verysmall part of it to make you believe: enter image description here Nevertheless, one can operate with it. For example,

Plot3D[int, {x, 0, 10}, {phiM, 0, 10}]

gives

enter image description here

Have fun!

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