Integrate[(Sin[n t/2]/Sin[t/2])^2, {t, 0, Pi}]~Table~{n, 10} // Timing

{2.17188, {π, 2π, 3π, 4π, 5π, 6π, 7π, 8π, 9π, 10π}}

Integrate[(Sin[n t/2]/Sin[t/2])^2, {t, 0, Pi},
Assumptions -> n ∈ Integers] // Timing
%[[-1]] ~ FunctionExpand ~ (Assumptions -> n ∈ Integers)
% // FullSimplify

-(1/2) I E^(-I n π) (1 + E^(2 I n π) +
2 E^(I n π) n π Cot[n π] + n PolyGamma[0, 1/2 - n/2] -
n PolyGamma[0, -(n/2)] + E^(2 I n π) n PolyGamma[0, n/2] -
E^(2 I n π) n PolyGamma[0, (1 + n)/2])

f[e_] := Count[e, _PolyGamma, Infinity]
Integrate[(Sin[n t/2]/Sin[t/2])^2 // TrigReduce, {t, 0, Pi}]
FullSimplify[%, ComplexityFunction -> f]
FullSimplify[%, n ∈ Integers]

$Version

(* "11.2.0 for Mac OS X x86 (64-bit) (September 11, 2017)" *)

Integrate[(Sin[n t/2]/Sin[t/2])^2, {t, 0, Pi}]~
  Table~{n, -5, 5} // AbsoluteTiming

(* {0.684515, {5 π, 4 π, 3 π, 2 π, π, 0, π, 2 π, 3 π, 4 π, 5 π}} *)

(int[n_] =
   Integrate[(Sin[n t/2]/Sin[t/2])^2 // TrigReduce, {t, 0, Pi}] //
       ComplexExpand // FullSimplify // FunctionExpand //
    FullSimplify) // AbsoluteTiming

(* {12.12901, (1/2)*(2 + n*Pi*Cot[(n*Pi)/2] -
        n*(PolyGamma[0, 1/2 - n/2] - 2*PolyGamma[0, n/2] +
             PolyGamma[0, (1 + n)/2]))*Sin[n*Pi]} *)

int[1]

(* Infinity::indet: Indeterminate expression 0 ComplexInfinity encountered.

Indeterminate *)

Table[Limit[int[n], n -> m], {m, -5, 5}]

(* {5 π, 4 π, 3 π, 2 π, π, 0, π, 2 π, 3 π, 4 π, 5 π} *)

(int[n_] = Integrate[(Sin[n t/2]/Sin[t/2])^2 // TrigReduce, {t, 0, Pi},
                      Assumptions -> n [Element] Integers && n > 0]) // Timing

(*     {10.094, -(1/2)
   I E^(-I n [Pi]) (1 + 2 E^(I n [Pi]) n [Pi] Cot[n [Pi]] +
   n PolyGamma[0, 1/2 - n/2] - n PolyGamma[0, -(n/2)] +
   E^(2 I n [Pi]) (1 + n PolyGamma[0, n/2] -
   n PolyGamma[0, (1 + n)/2]))}     *)

Table[Limit[int[n], n -> j] // FullSimplify, {j, 0, 10}]

(*     {0, Pi, 2 Pi, 3 Pi, 4 Pi, 5 Pi, 6 Pi, 7 Pi,
         8 Pi, 9 Pi, 10 Pi}     *)