Return result not in terms of hypergeometric functions

Question

I'm trying to sum over

summand = (2^(-2 + n) (1 - t)^(-1 + n/2) t^(
 1/2 (-1 + n)) η^n (1 + t η^2)^(-n/2)
  Gamma[-(1/2) + n] Gamma[1/2 + n]^2)/(π n! Gamma[-(1/2) + 2 n]) 

from n = 1 to infinity. I do this with the command

sum = Sum[summand, {n, 1, ∞}]

This returns a hypergeometric function

(-1 + HypergeometricPFQ[{-(1/2), 1/2, 1/2}, {-(1/4), 1/4}, (
  Sqrt[1 - t] Sqrt[t] η)/(
  2 Sqrt[1 + t η^2])])/(4 (1 - t) Sqrt[t])

I'm wondering whether it's possible to do this sum without getting a hypergeometric function (similar to ExcludedForms for Simplify).

Any help is appreciated.