The screenshot below shows a step of simplification that, intuitively, I thought made sense. Simply rewriting the one square root as a product of the sqrt(-1) times the remaining square root. Then writing the sqrt(-1) as i, and making it a coefficient.
$I = \displaystyle\frac{1}{\sqrt{i}}\arctan{\left(\sqrt{-i\tan{x}}\right)} - \frac{1}{\sqrt{i}}\operatorname{arctanh}{\left(\sqrt{-i\tan{x}}\right)} $
$I = \displaystyle\frac{1}{\sqrt{i}}\arctan{\left(i\sqrt{i\tan{x}}\right)} - \frac{1}{\sqrt{i}}\operatorname{arctanh}{\left(i\sqrt{i\tan{x}}\right)} $
However, upon investigating the graphs, these two answers are different. Can anyone explain the discrepancy? Did I make an error in assumption?