Using `NIntegrate` or `Integrate` with unspecified constants

Question

I have a function defined like :

g[n_, x_] := ((n + 2)*Cos[π*(n + 2)*x] - Cot[π*x] *Sin[(n + 2)*π*x])

and I want to do the following :

Integrate[x*g[n, x]*g[m, x], {x, 0, 1}]

But this results in the error :

Integrate::idiv: Integral of x ((2+m) Cos[(2+m) π x]-Cot[π x] Sin[(2+m) π x]) ((2+n) Cos[(2+n) π x]-Cot[π x] Sin[(2+n) π x]) does not converge on {0,1}.

On the other hand, if I specify values of n and m it gives me a numerical output, as desired. Experimenting with the numbers, I have been able to determine that whenever m = n + odd#, where odd# is any odd number > 1, the integral is zero within machine precision.

Table[Integrate[x*g[n, x]*g[n + i, x], {x, 0, 1}], {n, 3}, {i, 3}]

Results in :

{{-(6656/(735*Pi^2)), 0, -(2816/(2835*Pi^2))}, {-(15872/(945*Pi^2)), 0, -(72704/(38115*Pi^2))}, {-(1013504/(38115*Pi^2)), 0, -(1801216/(585585*Pi^2))}}

Considering the above, can someone please guide me how I can obtain the output of Integrate as a function of n or n and m? One way I can think of is to find the relation between coefficients of my output but I can't figure out how to do that by hand or by any code.

Edit : Inspired from comments I tried the following to check for specific cases of n and m values. Here I take m=n+3 which I have verified works for n upto 10 (atleast).

Integrate[x*g[n, x]*g[n + 3, x], {x, 0, 1}, 
 Assumptions -> {n \[Element] {1, 2, 3}}] (* Fails *)

Integrate[x*g[n, x]*g[n + 3, x], {x, 0, 1}] /. n -> 1 (* Works *)
Integrate[x*g[n, x]*g[n + 3, x], {x, 0, 1}] /. n -> 2 (* Works *)
Integrate[x*g[n, x]*g[n + 3, x], {x, 0, 1}] /. n -> 3 (* Works *)

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