I am studying the result of a Feynman Diagram, trying to apply the Renormalization Group to a Field theory for Active Matter. I have arrived at a point where I want to integrate with Mathematica a complicated expression, and I would need an analytical result, rather than a numerical one.
Using Integrate
, the code has been stuck on "running" for a day and it does not give a result nor an error of any kind. Does it mean that there is not an exact solution or does it mean something else?
Is there something I can try to get a result?
The code with the function I need to integrate is the following:
Integrand = (64 a w Cos[θ]^2 (f^3 κ^2 (w + κ +
2 w κ) λx0^5 +
4 w^3 (1 + w) Λ^6 λx0^5 +
2 w^2 (1 + w) Λ^6 λx0^4 (λx0 -
h λx0) +
f^3 κ^2 λx0^5 (-ht (1 + w) - κ -
w κ + h (1 + κ) + w μy +
w κ μy) +
f^3 κ^2 λx0^5 (-1 + μy) (-ht (2 + w) -
3 κ - 2 w κ + h (2 + 3 κ) + w μy +
2 w κ μy) Cos[θ]^2 +
f^2 κ Λ^2 λx0^5 (2 (-1 +
ht) w^3 - 2 κ (-2 + h + μy) +
w^2 (-15 + 9 ht - 18 κ + h (4 + 6 κ) +
2 μy + 12 κ μy) +
w (-4 + 4 ht +
5 κ (-2 + h + μy))) Cos[θ]^2 +
f^2 κ Λ^2 λx0^4 (2 κ ( λx0 - h λx0) (-2 + h + μy) -
w^2 λx0 (h + 3 μy - 4 h μy -
2 κ (-1 + μy) (-3 + 2 h + μy) +
ht (-4 + 3 h + μy)) +
w λx0 (6 κ + h^2 (2 + κ) -
2 ht (-3 + μy) - 2 μy -
6 κ μy + κ μy^2 +
h (-4 - 4 ht - 6 κ + 4 μy +
4 κ μy))) Cos[θ]^2 -
f^2 (-1 + h) κ Λ^2 λx0^5 (6 κ +
h^2 κ - 6 w κ - 3 w^2 κ +
h w (2 + ht - 3 κ (-1 + μy) - 3 μy) +
w μy - 6 κ μy + 9 w κ μy +
6 w^2 κ μy + κ μy^2 -
3 w κ μy^2 - 3 w^2 κ μy^2 +
ht w (-3 + 2 μy) +
2 h κ (-3 + 2 μy)) Cos[θ]^4 +
f (-1 + h) w Λ^4 λx0^5 (2 (-1 +
ht) w^3 - 8 κ (-1 + μy) +
w (-12 + 10 ht - 27 κ + h (2 + 20 κ) +
7 κ μy) +
w^2 (-18 + 15 ht - 20 κ + h (3 + 8 κ) +
12 κ μy)) Cos[θ]^4 +
f (-1 + h)^2 w Λ^4 λx0^5 (-2 ht w -
2 h (6 κ +
w (-1 + 2 κ)) + κ (-6 (-3 + μy) +
2 w^2 (-1 + μy) + w (3 + μy))) Cos[θ]^4 -
2 (-1 + h)^4 w^2 (4 +
w) Λ^6 λx0^5 Cos[θ]^6 +
2 (-1 + h)^3 w^2 (6 + 11 w +
2 w^2) Λ^6 λx0^5 Cos[θ]^6 -
f (-1 + h)^3 w κ Λ^4 λx0^5 (-12 + 4 h + w (-1 + μy) + 8 μy) Cos[θ]^6 + 2 w^2 Λ^6 (λx0 - h λx0)^5 Cos[θ]^8 - f^2 κ^2 λx0^4 (λx0 - h λx0) (-1 + μy)^2 Cos[θ]^6 (f κ (-1 + μy) - (-1 + h)^2 Λ^2 Cos[θ]^2) - w Λ^4 λx0^5 (-f (1 + w) (-ht w (2 + w) + h (2 w + w^2 - 4 κ) + κ (3 + 2 w (-1 + μy) + μy)) + 2 (-1 + h)^2 w (4 + 3 w) Λ^2 Cos[θ]^2) + 2 w Λ^4 λx0^5 (f (w + w^3 - κ + 2 w κ + w^2 (3 + 4 κ)) + (-1 + h) w (1 + 9 w + 6 w^2) Λ^2 Cos[θ]^2) - f κ λx0^5 (1 - μy) Cos[θ]^6 (f^2 κ (1 - μy) (h + w - ht (1 + w) + κ + w κ - κ μy - w κ μy) +f (-1 + h)^2 Λ^2 (κ + w (-1 + ht + κ (-1 + μy)) - κ μy) Cos[θ]^2 + (-1 + h)^4 w Λ^4 Cos[θ]^4) + f κ λx0^5 (1 - μy) Cos[θ]^4 (f^2 κ (h - ht + 3 h κ + κ (-3 + w (-1 + μy))) (1 -μy) + 2 f (-1 + h)^2 κ Λ^2 (-2 + h + w + μy - w μy) Cos[θ]^2 + 3 (-1 + h)^4 w Λ^4 Cos[θ]^4) + Λ^2 λx0^5 (f^2 κ (-ht w (3 + 4 w + w^2) + κ - 2 w κ - 3 w^2 κ + h (-κ + w^2 (1 + κ) + w (2 + κ)) + w μy + 3 w^2 μy + w^3 μy + w κ μy + 2 w^2 κ μy) + f (-1 + h) w Λ^2 (-ht w (4 + 3 w) + h (3 w^2 + w (4 - 8 κ) - 12 κ) + κ (12 + 4 w^2 (-1 + μy) + w (3 + 5 μy))) Cos[θ]^2 - 6 (-1 + h)^3 w^2 (2 +w) Λ^4 Cos[θ]^4) + Λ^2 λx0^5 (f^2 κ (w + w^3 - κ + 3 w κ + w^2 (5 + 7 κ)) + f w Λ^2 ((-4 + h + 3 ht) w^3 + κ (7 - 4 h - 3 μy) + w (-8 + 4 ht - 17 κ + 4 h (1 + 4 κ) + κ μy) + w^2 (-18 + 9 ht - 22 κ + h (9 + 16 κ) + 6 κ μy)) Cos[θ]^2 + 2 (-1 + h)^2 w^2 (4 + 15 w + 6 w^2) Λ^4 Cos[θ]^4) - (1/λtx0)λx0^5 Cos[θ]^2 (-f^3 κ^2 λtx0 (-h + ht - 3 w + 2 ht w - 3 κ - 5 w κ + w μy + 3 κ μy + 5 w κ μy) + f^2 κ Λ^2 (w^3 (λtx0 - ht λtx0) (-1 + μy) - w^2 λtx0 (15 + h (-9 + μy) - 7 μy + 5 κ (-1 + μy) (-3 + 2 h + μy) + 2 ht (-7 + 4 h + 3 μy)) + κ λtx0 (6 + h^2 - 6 μy + μy^2 + h (-6 + 4 μy)) - w λtx0 (6 + 12 κ + h^2 (-1 + 2 κ) + 3 ht (-3 + μy) - μy - 12 κ μy + 2 κ μy^2 + 2 h (-1 + 3 ht - 6 κ - μy + 4 κ μy))) Cos[θ]^2 - f (-1 + h)^2 w Λ^4 λtx0 (6 w^2 (-1 + ht + κ (-1 + μy)) + 2 κ (1 + 2 h - 3 μy) + w (-8 + 8 ht + κ (-19 + 8 h + 11 μy))) Cos[θ]^4 - 2 (-1 + h)^4 w^2 (4 + 3 w) Λ^6 λtx0 Cos[θ]^6) + λx0^4 Cos[θ]^4 (f^3 κ^2 λx0 (-1 + μy) (-2 h - 3 w + ht (2 + 3 w) - 3 κ - 4 w κ + 3 κ μy + 4 w κ μy) + f^2 κ Λ^2 (λx0 - h λx0) (w^2 (5 - 5 ht - 4 κ (-1 + μy)) (-1 + μy) + 2 κ (-1 + μy) (-2 + h + μy) + w (-3 κ (-1 + μy) (-2 + h + μy) - 2 ht (-3 + h + 2 μy) + 2 (-2 + μy + h μy))) Cos[θ]^2 + f (-1 + h)^3 w Λ^4 λx0 (2 (-1 +
h) κ +
w (-2 + 2 ht +
5 κ (-1 + μy))) Cos[θ]^4 +
2 (-1 + h)^5 w^2 Λ^6 λx0 Cos[θ]^6)) Sin[θ]^2)/(h π λx0^5 (1 + h + 2 w + (-1 + h) Cos[2 θ])^3 ((1 + h) w Λ^2 + f κ (1 + μy) + ((-1 + h) w Λ^2 + f κ (-1 + μy)) Cos[2 θ])^3);
Integrate[Integrand, {θ, 0, π},
Assumptions -> {w > 0, h > 0, h != 1, μy != 1, μy > 0, ht > 0, Λ > 0, κ > 0}]
Integrand
doesn't evaluate, please check your code! $\endgroup$Integrate
would take a long time. My advice here is to think about why you need this integral in symbolic form. Symbolic forms are primarily useful for humans (as opposed to computers) to understand. The integrand you have here is already too large for that. Suppose you had a result: what would you do with it, how does it get you closer to your actual goal? $\endgroup$