Properties of $f(x)=\ln(1+x^2)+x+2$ vs $g(x)=\cosh x+\sinh x$ This is from an MCQ contest.

Consider the two functions: $f(x)=\ln(1+x^2)+x+2$ et $g(x)=ch(x)+sh(x)$.
The real number $c$ such that: $(f^{-1})'(2)=g(c)$
  
  
*
  
*$1]$ $c=-1$
  
*$2]$ $c=0$
  
*$3]$ $c=1$
  
*$4]$ None of the above statements is correct
  

My Thoughts:
note that :
$$\left(f^{-1}\right)^\prime (y)=\frac{1}{f'\left(f^{-1}(y)\right)}.$$
and 
$$ch(x)+sh(x)=e^x$$
but first we should check $f$ it's invertible or not 
indeed,
$f$ is strictly increasing continuous on $\mathbb{R}$ since 
$$f'(x)=\dfrac{(x+1)^{2}}{x^2+1}\geq 0$$ 
then $f$ is invertible thus has inverse function
$$f(x)=y\iff x=f^{-1}(x) \\
\ln(1+x^2)+x+2=y
$$
i'm stuck here 
Any help will be apprecited

$$f^{-1}(2)=t$$
$$f(t)=2$$
$$\ln(1+t^2)+t+2=2$$
Should I continue?
