Is this a known distribution similar to normal distribution with a variable variance?

Question

I want to know that is there any distribution with a pdf of the form $$ f(x) = c \,(1+x^2)^{-1/2}\exp\left(-\frac{1}{2(1+x^2)}(x-a)^2\right)$$?

Answer 1

Your function is not integrable so it is not a density function. Note that $f(x) \to ce^{-1/2}$ as $x \to \infty $.