Symbolic integral not computed

FWIW, Wolfram|Alpha can denest this radical:

radical = WolframAlpha[
  "Simplify[Sqrt[Sqrt[1 + x^2] - 1]]", 
  {{"Result", 1}, "ComputableData"}
][[1]]

Integrate[(Sqrt[x] - radical) Sin[x], {x, 0, ∞}]

((-1 + E) Sqrt[\[Pi]/2])/(2 E)

FWIW, you can de-nest roots using the identity

$$\sqrt{\frac{1}{2} (-a+i b x)}+\sqrt{\frac{1}{2} (-a-i b x)}=\sqrt{\sqrt{a^2+b^2 x^2}-a}$$ from which it follows that $$ \int_{\mathbb R^+}\sqrt{\sqrt{a^2+b^2 x^2}-a}\ \sin x\mathrm dx=\frac{1}{2} \sqrt{\frac{\pi }{2}} \sqrt{b} \left(1-e^{-\frac{a}{b}}\right) $$ as given by

Integrate[
          (Sqrt[b x] - Sqrt[(-a + I x b)/2] - Sqrt[(-a - I x b)/2]) Sin[x]
         , {x, 0, Infinity}, Assumptions -> a > 0 && b > 0
]

Your integral corresponds to $a=b=1$,

% /. {a -> 1, b -> 1}
(* 1/2 (1 - 1/E) Sqrt[Pi/2] *)

as expected.