Series command does not work

Question

 Series[ Exp[Sin [Sqrt[1 + x^2]]], {x, Infinity, 1}]

E^Sin[Sqrt[1 + x^2]]

Output is same as input whereas I want this answer...

E^Sin[1/(2 x) + x + ...]

WITHOUT having to manually expand the square root separately. I thinking of a long expression where many Exponentials and Sines and square roots are present in that expression.

Answer 1

Try this:

MapAt[Normal[Series[#, {x, \[Infinity], 1}]] &, expr, {2, 1}]

(*  E^Sin[1/(2 x) + x]  *)

Have fun!