Sorry Roberto, but you've made a mistake here.

The generators of rotation around the 3 axes are the matrices

, and therefore a finite rotation through

is given by the exponentiation:

So it's not in fact the Pauli matrices

themselves we have to exponentiate (as the text would seem to indicate), but the

. (Or

for rotation about an arbitrary axis, using your notation).

You've included (without really explaining why) the factor of

, but forgotten the

. Hence the bracketed terms in your expansions of

and

are

not in fact equal to

and

as you have indicated. Recall that the power series for

sin and

cos both have alternating + and - terms; that isn't the case here!!!

My solution (which is really just the same as yours but with these problems addressed) is attached below.