Never ending Integrate computation

Question

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}]

Answer 1

the expression probably integrates, but it will take a long time. I expanded it and got about 440 individual terms where each one (I picked random ones to test) will take about 10 mins to give a symbolic result on a quite fast computer. So it may take over 70 hours in total and give a VERY large expression...

Edit:

I found a way to get at least to the Antiderivate in reasonable time:

After expansion of the integrand collect and simplify all terms of equal theta dependence. After that 6 different integrals remain which can be done in about 2000 secs. The result and the calculation of the definite integral should be checked numerically by giving numerical values to the numerous parameters. The antiderivative contains terms with

ArcTan[(Sqrt[f*\[Kappa] + w*\[CapitalLambda]^2]*Tan[\[Theta]])/
Sqrt[h*w*\[CapitalLambda]^2 + f*\[Kappa]*\[Mu]y]]

which are zero for both theta equal 0 or pi. And it is not clear whether the antiderivative is continuous within the integration range.

If the limits are plugged in without check the following expression remains:

(1/(2*(-1 + h)*
 h*((-1 + h)*w*\[CapitalLambda]^2 + 
    f*\[Kappa]*(-1 + \[Mu]y))^4))*(a*
w*\[CapitalLambda]^2*(2*(-1 + h)^3*(1 + 3*h)*
  w^3*\[CapitalLambda]^6 - 
 2*f^3*\[Kappa]^2*(w*(-1 + ht + \[Kappa]*(-1 + \[Mu]y)) - 
    h*\[Kappa]*(-1 + \[Mu]y))*(-1 + \[Mu]y)^2 - 
     
 f^2*(-1 + h)*
  w*\[Kappa]*\[CapitalLambda]^2*(-1 + \[Mu]y)*(6*
     w*(-1 + ht + \[Kappa]*(-1 + \[Mu]y)) + \[Kappa]*(1 + 
       7*h + \[Mu]y - 9*h*\[Mu]y)) + 
 f*(-1 + h)^2*
  w^2*\[CapitalLambda]^4*(-4*
     w*(-1 + ht + \[Kappa]*(-1 + \[Mu]y)) + \[Kappa]*(-3 - 5*h - 
       5*\[Mu]y + 13*h*\[Mu]y))))