As I said

Please make sure I have fixed them correctly.

Did you verify that the expressions are correctly parenthesized and Sqrt5 should be Sqrt[5]?

I did find something interesting. If I first evaluate

Integrate[lhs, {x, -2, 2}]

and then evaluate

Integrate[lhs, {x, 0, xSol}]

The result is a simpler expression that is numerically equivalent

(* 1/240 (-3 Sqrt[5 (35 - 13 Sqrt[5] + Sqrt[30 (65 - 29 Sqrt[5])])] + 8 π) *)

This odd behavior is reproducible (at least for me on "12.3.1 for Mac OS X x86 (64-bit) (June 19, 2021)"). I will report it to Wolfram support.

There are some misplaced parenthesis and missing [ ]. Please make sure I have fixed them correctly.

lhs = -1/4 Sqrt[2 - 2/Sqrt[5]] + Sqrt[1 - (x + 1/4 (1 + Sqrt[5]))^2]
rhs = x Sqrt[1 + 2/Sqrt[5]]
eqn = lhs == rhs

sol = Solve[eqn, x]
xSol = x /. First@sol // Simplify
(* 1/32 (-4 Sqrt[5] + Sqrt[2 (3 + Sqrt[5])] - Sqrt[10 (3 + Sqrt[5])] + 
   2 Sqrt[2 (60 - 16 Sqrt[5] + 5 Sqrt[2 (3 + Sqrt[5])] - Sqrt[10 (3 + Sqrt[5])])]) *)

Integrate[lhs, {x, 0, xSol}]
(* 1/32 Sqrt[10 - 2 Sqrt[5]] - 1/32 Sqrt[10 - 2 Sqrt[5]] (1 + Sqrt[5]) - 
 1/32 Sqrt[1/10 (5 + Sqrt[5])] + 1/32 Sqrt[1/2 (5 + Sqrt[5])] - 1/(
 16 Sqrt[-(
   2/(-38 + 14 Sqrt[5] - 3 Sqrt[2 (3 + Sqrt[5])] + Sqrt[
    10 (3 + Sqrt[5])]))]) + (1/2048)(8 + 4 Sqrt[5] + Sqrt[
    2 (3 + Sqrt[5])] - Sqrt[10 (3 + Sqrt[5])] + 
    2 Sqrt[
     2 (60 - 16 Sqrt[5] + 5 Sqrt[2 (3 + Sqrt[5])] - Sqrt[
        10 (3 + Sqrt[5])])]) \[Sqrt]((24 - 4 Sqrt[5] - Sqrt[
       2 (3 + Sqrt[5])] + Sqrt[10 (3 + Sqrt[5])] - 
       2 Sqrt[
        2 (60 - 16 Sqrt[5] + 5 Sqrt[2 (3 + Sqrt[5])] - Sqrt[
           10 (3 + Sqrt[5])])]) (40 + 4 Sqrt[5] + Sqrt[
       2 (3 + Sqrt[5])] - Sqrt[10 (3 + Sqrt[5])] + 
       2 Sqrt[2 (60 - 16 Sqrt[5] + 5 Sqrt[2 (3 + Sqrt[5])] - Sqrt[10 (3 + Sqrt[5])])])) - 1/2 I Log[8] - 
 1/2 I Log[1 + Sqrt[5] + I Sqrt[10 - 2 Sqrt[5]]] + 
 1/2 I Log[
   8 + 4 Sqrt[5] + Sqrt[2 (3 + Sqrt[5])] - Sqrt[10 (3 + Sqrt[5])] + 
    2 Sqrt[
     2 (60 - 16 Sqrt[5] + 5 Sqrt[2 (3 + Sqrt[5])] - Sqrt[10 (3 + Sqrt[5])])] + 
    2 I \[Sqrt](2 (48 + 8 Sqrt[5] - 2 Sqrt[2 (3 + Sqrt[5])] + 
          2 Sqrt[10 (3 + Sqrt[5])] - 
          4 Sqrt[2 (60 - 16 Sqrt[5] + 5 Sqrt[2 (3 + Sqrt[5])] - Sqrt[10 (3 + Sqrt[5])])] - 
          2 Sqrt[10 (60 - 16 Sqrt[5] + 5 Sqrt[2 (3 + Sqrt[5])] - Sqrt[
              10 (3 + Sqrt[5])])] - Sqrt[2 (50 + 6 Sqrt[5] + 5 Sqrt[2 (3 + Sqrt[5])] + Sqrt[
             10 (3 + Sqrt[5])])] + Sqrt[10 (50 + 6 Sqrt[5] + 5 Sqrt[2 (3 + Sqrt[5])] + Sqrt[
             10 (3 + Sqrt[5])])]))] *)