Can Mathematica solve $\int \frac{\cot^2\left(\frac12\sec^{-1}(x)\right)}{\sqrt{\tan\left(\frac12\csc^{-1}(x)\right)}} \, dx$?

Can Mathematica solve integral $(1)$?

$$\int \frac{\cot^2\left(\frac12\sec^{-1}(x)\right)}{\sqrt{\tan\left(\frac12\csc^{-1}(x)\right)}} \, dx.\tag{1}$$

This is

Integrate[Cot[1/2 ArcSec[x]]^2 / Sqrt[Tan[1/2 ArcCsc[x]]], x]

Perhaps it is worth clarifying that it is not that I do not know how to solve it. The integral can be mechanically solved using the method described in this blog post. But I have noticed that both Wolfram Alpha and this Integral Calculator are unable to simplify it, let alone solve it (I mean they do not provide the closed form of the integral).