Is composition of analytic functions itself analytic? Is composition of analytic functions itself analytic?

Is there a proof that, say,

$$f(x)=e^{\frac{x^2+1}{x^2-1}}$$

analytic?
The composition of analytic functions indeed is analytic. The fastest proof surely relies on complex analysis: every analytic function of one real variable is the restriction of a holomorphic function of one complex variable, so the statement is a consequence of the (complex variable) chain rule.
Your example is a composition/combination of several functions
$$
u(x) = e^x \\
v(x) = x^2 + 1\\
h(x) = x^2  - 1 \\
m(x, y) = x / y \\
f(x) = u( m( v(x), h(x) ) )
$$
The function $m$ is not very nice when $y = 0$, but $y = 0$ happens to be in the image of $h$. 

What are you assuming is the domain of $f$? And are you asking "Is it analytic on that domain?"?