inverse Gudermannian function

Since the real Gudermannian function gd is strictly increasing and forms a bijection from $ℝ$ onto the open interval  $(-\frac{\pi }{2},\frac{\pi }{2})$,  it has an inverse function

$${\text{gd}}^{-1}:(-\frac{\pi }{2},\frac{\pi }{2})\to ℝ.$$

The function ${\text{gd}}^{-1}$ is denoted also arcgd.

If  $x=\text{gd}y$, which may be explicitly written e.g.

$$x=\arcsin (\tanh y),$$

one can solve this for $y$, getting first  $\tanh y=\sin x$  and then

$$y=\text{artanh}(\sin x)$$

(see the area functions).  Hence the inverse Gudermannian is expressed as

${\text{gd}}^{-1}(x)=\text{arcgd}x=\text{artanh}(\sin x)$(1)

It has other equivalent (http://planetmath.org/Equivalent3) expressions, such as

${\text{gd}}^{-1}(x)=\text{arsinh}(\tan x)={\displaystyle \frac{1}{2}}\ln {\displaystyle \frac{1+\sin x}{1-\sin x}}={\displaystyle {\int }_0^x}{\displaystyle \frac{dt}{\cos t}}.$(2)

Thus its derivative is

${\displaystyle \frac{d}{dx}}{\text{gd}}^{-1}(x)={\displaystyle \frac{1}{\cos x}}.$(3)

Cf. the formulae (1)–(3) with the corresponding ones of gd.