Arctangent of Imaginary Number

Theorem
Let $x$ be a real number.

Then:
 * $\tan^{-1} \left({ix}\right) = \dfrac i 2 \ln \left({ \dfrac {1 + x} {1 - x} }\right)$

where $\tan$ is the complex tangent function, $\ln$ is the real natural logarithm, and $i$ is the imaginary unit.

Proof
Let $y = \tan^{-1} \left({ix}\right)$.

Let $x = \tanh \theta$, then $\theta = \tanh^{-1} x$.