Inconsistent result of definite integration vs indefinite integration

Question

My question pertains to the inconsistency that may arise in indefinite integrals vs definite integrals in Mathematica due to the indefinite integral being discontinuous as explained here.

I want to evaluate the following integral and obtain the result as a function of n and m, the latter is absolutely crucial to me :

$ g(n,x) = \sqrt{\frac{2}{(n+2)^2 - 1}} \left\{ (n+2) \cos[(n+2) \pi x] - \cot(\pi x) \sin[ (n+2) \pi x] \right\}$ and $x_{nm}(n,m) = \int_0^1 dx\ g(n,x)\ x\ g(m,x)$.

Naively, integrating this results in a non-converging integral. So, I tried to do an indefinite integral and take the limits and then subtract to get the result. This gives me inconsistent result when n=m and when |n-m| is even. In both cases I get indeterminate result whereas if I go with definite integral (while specifying the value of n and m) then in the former I get finite result and in the latter I get 0. Except this the method works absolutely fine.

I realize that this comes from the discontinuity in the anti-derivative obtained using Integrate and I have read the blog post linked above but can't make it work for me. Can someone please help me resolve the inconsistency?

All Mathematica Code pertaining to this is below :-

g[n_, x_] := 
  Sqrt[2/((n + 2)^2 - 
    1)] ((n + 2)*Cos[\[Pi]*(n + 2)*x] - 
     Cot[\[Pi]*x]*Sin[(n + 2)*\[Pi]*x]);

Integrate[ g[n, x] * g[m, x] *x, {x, 0, 1}, 
 Assumptions -> {n \[Element] Integers, m \[Element] Integers}] (* Indeterminate *)

Integrate[ g[n, x] * g[m, x] *x, x, 
 Assumptions -> {n \[Element] Integers, m \[Element] Integers}]

(* Output-

2 Sqrt[1/(3 + 4 m + m^2)] Sqrt[1/(
 3 + 4 n + 
  n^2)] (-(1/(
    2 (m - n) (4 + m + n) \[Pi]))((
      2 (2 + m) (3 + 4 m + m^2) (2 + 
         n) Cos[(2 + m) \[Pi] x] Cos[(2 + n) \[Pi] x])/((-m + n) (4 + 
         m + n) \[Pi]) - (1/(
      2 \[Pi]))(-((
         E^(I (m - n) \[Pi] x) (m^2 - m (-1 + 4 n + n^2) - 
            3 (2 + 4 n + n^2)) Hypergeometric2F1[1, (m - n)/2, 
           1/2 (2 + m - n), E^(2 I \[Pi] x)])/(m - n)) - (
        E^(I (2 + m - n) \[Pi] x) (6 + m^2 + 4 n + n^2 + 
           m (7 + 4 n + n^2)) Hypergeometric2F1[1, 1/2 (2 + m - n), 
          1/2 (4 + m - n), E^(2 I \[Pi] x)])/(2 + m - n) + (
        1/((2 + m + n) (4 + m + n)))
        E^(-2 I (3 + m + 
            n) \[Pi] x) (-E^(
             I (2 + m + n) \[Pi] x) (12 + m^3 + 14 n + 6 n^2 + n^3 + 
              m^2 (9 + 5 n + n^2) + 
              m (20 + 19 n + 7 n^2 + n^3)) Hypergeometric2F1[
             1, -2 - m/2 - n/2, -1 - m/2 - n/2, E^(2 I \[Pi] x)] - 
           E^(I (4 + m + n) \[Pi] x) (m^3 - m^2 (-5 + 3 n + n^2) - 
              3 (8 + 18 n + 8 n^2 + n^3) - 
              m (2 + 27 n + 11 n^2 + n^3)) Hypergeometric2F1[
             1, -1 - m/2 - n/2, -(m/2) - n/2, E^(2 I \[Pi] x)])) - (
      1/(2 \[Pi]))((
        E^(-I (m - n) \[Pi] x) (6 + m^2 + 4 n + n^2 + 
           m (7 + 4 n + n^2)) Hypergeometric2F1[1, 1/2 (-m + n), 
          1/2 (2 - m + n), E^(2 I \[Pi] x)])/(m - n) + (
        E^(-I (-2 + m - n) \[Pi] x) (m^2 - m (-1 + 4 n + n^2) - 
           3 (2 + 4 n + n^2)) Hypergeometric2F1[1, 1/2 (2 - m + n), 
          1/2 (4 - m + n), E^(2 I \[Pi] x)])/(-2 + m - n) + (
        1/((4 + m + n) (6 + m + n)))(E^(
           I (4 + m + n) \[Pi] x) (m^3 - m^2 (-7 + 3 n + n^2) - 
             m n (35 + 13 n + n^2) - 
             3 (12 + 26 n + 10 n^2 + n^3)) Hypergeometric2F1[1, 
            1/2 (4 + m + n), 1/2 (6 + m + n), E^(2 I \[Pi] x)] + 
          E^(I (6 + m + n) \[Pi] x) (24 + m^3 + 22 n + 8 n^2 + n^3 + 
             m^2 (11 + 5 n + n^2) + 
             m (34 + 27 n + 9 n^2 + n^3)) Hypergeometric2F1[1, 
            1/2 (6 + m + n), 1/2 (8 + m + n), E^(2 I \[Pi] x)])) + (
      2 (3 + 4 m + m^2) (2 + 
         n)^2 Sin[(2 + m) \[Pi] x] Sin[(2 + n) \[Pi] x])/((-m + 
         n) (4 + m + n) \[Pi])) + (1/(2 (m - n) (4 + m + n) \[Pi]))
   x (2 (3 + 4 m + m^2) (2 + 
         n) Cos[(2 + n) \[Pi] x] Sin[(2 + m) \[Pi] x] - 
      2 ((2 + m) (3 + 4 n + n^2) Cos[(2 + m) \[Pi] x] + (4 m + m^2 - 
            n (4 + n)) Cot[\[Pi] x] Sin[(2 + m) \[Pi] x]) Sin[(2 + 
           n) \[Pi] x]))
*)

(* Taking limits, subtracting and using a FullSimplify gives the following *)

(1/(2 \[Pi]^2))(-1 + (-1)^(m + n)) Sqrt[1/(3 + 4 m + m^2)] Sqrt[1/(
 3 + 4 n + 
  n^2)] ((4 (2 + m) (2 + n) (6 + m (4 + m) + n (4 + n)))/((m - 
      n)^2 (4 + m + n)^2) + HarmonicNumber[1/2 (-2 + m - n)] - 
   HarmonicNumber[-3 - m/2 - n/2] + HarmonicNumber[1/2 (-2 - m + n)] -
    HarmonicNumber[1/2 (2 + m + n)])

(* Trying for m->n limit and checking the result : *)
Limit[1/(2 \[Pi]^2) (-1 + (-1)^(m + n)) Sqrt[1/(3 + 4 m + m^2)] Sqrt[
  1/(3 + 4 n + 
   n^2)] ((4 (2 + m) (2 + n) (6 + m (4 + m) + n (4 + n)))/((m - 
       n)^2 (4 + m + n)^2) + HarmonicNumber[1/2 (-2 + m - n)] - 
    HarmonicNumber[-3 - m/2 - n/2] + 
    HarmonicNumber[1/2 (-2 - m + n)] - 
    HarmonicNumber[1/2 (2 + m + n)]), m -> n]

(* Output : DirectedInfinity[Sign[-1 + (-1)^(2 n)]/Sqrt[1/
  Sign[3 + 4 n + n^2]]] Sqrt[1/Sign[3 + 4 n + n^2]] *)

With[{n = 2, m = 2}, Integrate[ g[n, x] * g[m, x] *x, {x, 0, 1}]]

(* Output : 1/2 *)

Please help me get rid of the inconsistency.

Edit :

I had asked a related question here. I could make things work using Assumptions as explained in the comments there and then got stuck at the above-mentioned inconsistency.