calculus and analysis – Useless result of Integrate

Considering a Fourier sine series, I calculate

int = Integrate(Log(1 + Sin(x))*Sin(i*x), {x, 0, Pi}, Assumptions -> i (Element) PositiveIntegers)

1/4 (-((2 E^(-(1/2) I i (Pi)))/i^2) + (2 E^((I i (Pi))/2))/i^2 + E^(-I i (Pi))/i^2 - E^(I i (Pi))/i^2 + ( 4 (E^((I i (Pi))/2) - Cos((i (Pi))/2)) HurwitzLerchPhi(-1, 1, 1 - i))/i - ( 2 I (1 - E^(-I i (Pi))) HurwitzLerchPhi(-I, 1, 1 - i))/i - ( 2 I LerchPhi(-I, 1, 1 + i))/i + ( 2 I E^(I i (Pi)) LerchPhi(-I, 1, 1 + i))/i - ( E^(-(1/2) I i (Pi)) (-1 + E^(I i (Pi))) PolyGamma(0, 1 + i/2))/ i + (E^(-(1/2) I i (Pi)) (-1 + E^(I i (Pi))) PolyGamma(0, ( 1 + i)/2))/i + ( 4 HypergeometricPFQ({1, 1, 3/2}, {2 - i/2, 2 + i/2}, 1) Sin(( i (Pi))/2)^2)/(-4 i + i^3) - ( 2 HypergeometricPFQ({1/2, 1, 1}, {3/2 - i/2, 3/2 + i/2}, 1) Sin( i (Pi)))/(-1 + i^2))

Unfortunately, the result is useless in view of

Table(int, {i, 1, 5})

{Indeterminate, Indeterminate, Indeterminate, Indeterminate, Indeterminate}

and

Limit(int, i -> 3)

which returns the input.

It should be noticed that

Table(Integrate(Log(1 + Sin(x))*Sin(i*x), {x, 0, Pi}), {i, 1, 5})

results in {-2 + (Pi), 0, 1/9 (10 - 3 (Pi)), 0, -(46/75) + (Pi)/5}.

Is there a workaround for the general case?