I am computing a Feynman Diagram, to apply the Renormalization Group to an active field theory. Doing it, I need 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?
The code with the function I need to integrate is the following (together with the Integrate command):
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}]
