Computing integrals in Mathematica

Question

I am trying to actually compute $\int \sqrt{\cosh{y}-\cos{x}}e^{inx} dx$ in mathematica. As an example i tried computing $\int \sqrt{1-\cos{x}}e^{inx} dx$ and got a result. But when i try $\int \sqrt{2-\cos{x}}e^{inx} dx$ i do not get any result. Note that $n\in \mathbb{Z}$. Any fixes please.

The code i used was:

Integrate $\bigg[\sqrt{2-Cos[x]} Exp[i*n*x],x\bigg]$

Integrate[Sqrt[2-Cos[x]]Exp[I n x],x]

Answer 1

The function cannot be integrated with unknown n, because its form changes with different values for n:

Table[Integrate[Sqrt[2 - Cos[x]] Exp[I n x], x], {n, 0, 3}]

1:

2 EllipticE[x/2, -2], 1/3 (-4 EllipticE[x/2, -2] + 6 EllipticF[x/2, -2] - I Sqrt[2 - Cos[x]] (-5 + 2 Cos[x/2]^2 + Cos[x] + 2 I Sin[x]))

2:

2/15 (I + Cot[x]) (13 Sqrt[3] Sqrt[-1 + Cos[x]] Sqrt[1 + Cos[x]] EllipticE[ArcSin[Sqrt[2 - Cos[x]]], 1/3] (Cos[x] - I Sin[x]) - 5 Sqrt[3] Sqrt[-1 + Cos[x]] Sqrt[1 + Cos[x]] EllipticF[ArcSin[Sqrt[2 - Cos[x]]], 1/3] (Cos[x] - I Sin[x]) + Sqrt[2 - Cos[x]] (8 - 8 Cos[2 x] + 4 I Sin[x] + 5 I Sin[2 x]))

3:

1/35 Csc[x] (Cos[(3 x)/2] + I Sin[(3 x)/2]) (132 Sqrt[3] Sqrt[-1 + Cos[x]] Sqrt[1 + Cos[x]] EllipticE[ArcSin[Sqrt[2 - Cos[x]]], 1/ 3] (Cos[(3 x)/2] - I Sin[(3 x)/2]) - 50 Sqrt[3] Sqrt[-1 + Cos[x]] Sqrt[1 + Cos[x]] EllipticF[ArcSin[Sqrt[2 - Cos[x]]], 1/ 3] (Cos[(3 x)/2] - I Sin[(3 x)/2]) + Sqrt[2 - Cos[x]] (85 Cos[x/2] - 14 Cos[(3 x)/2] - 71 Cos[(5 x)/2] - 39 I Sin[x/2] + 22 I Sin[(3 x)/2] + 61 I Sin[(5 x)/2]))