Arctangent Function in terms of Gaussian Hypergeometric Function

Theorem

 * $\arctan x = x \map F {\dfrac 1 2, 1; \dfrac 3 2; -x^2}$

where:
 * $x$ is a real number with $\size x < 1$
 * $\arctan$ denotes the arctangent function
 * $F$ denotes the Gaussian hypergeometric function.

Also presented as
This result can also be seen presented in the form:


 * $\map F {\dfrac 1 2, 1; \dfrac 3 2; -x^2} = \dfrac {\arctan x} x$