Find analytic solution for integral only defined for even integers

Question

I would like to calculate the integral $$\int_0^{2\pi} dx \sin^6\left(\frac{x}{2}\right) F\left(\frac{4-n}{2}, \frac{4+n}{2}, \frac{1}{2}, \cos^2 \frac{x}{2} \right)^2$$ where $F$ is the hypergeometric function (Hypergeometric2F1 in Mathematica). One can see in Mathematica that this integral has some well-defined analytic solution when $n$ is an even integer, excluding $0, -2, 2$ (this is related to the fact that the hypergeometric function is a simple polynomial in these cases). For example, $n=4$ gives $5\pi/8$ and $n=6$ gives $45\pi/32$. I would like to obtain an analytic expression for this integral - however, Mathematica obviously cannot solve this for general $n$. I've tried using Assumptions in the integral, but Mathematica still cannot solve the expression for variable $n$.

Answer 1

A = Table[{n, 
      Integrate[Sin[x/2]^6 Hypergeometric2F1[(4-n)/2, (4+n)/2, 1/2, Cos[x/2]^2]^2,
                {x, 0, 2π}]}, {n, 4, 20, 2}]

(*    {{4, 5π/8}, {6, 45π/32}, {8, 33π/16}, {10, 429π/160},
       {12, 1287π/392}, {14, 12155π/3136}, {16, 20995π/4704},
       {18, 323π/64}, {20, 7429π/1320}}                          *)

FindSequenceFunction[A, n] // FullSimplify

(*    (2 (9 - 10 n^2 + n^4) π)/(7 n (-4 + n^2))    *)