First step for $\int\frac{\operatorname{gd}\left(\log x\right)}{\sqrt{1-x^2}}dx$, where $\operatorname{gd}(x)$ denotes the Gudermannian function

Question

In this post we denote the Gudermannian function as $\operatorname{gd}(x)$, I am saying the function from this Wikipedia.

I believe that it is possible to get the indefinite integral of $$\int\frac{\operatorname{gd}\left(\log x\right)}{\sqrt{1-x^2}}dx.\tag{1}$$

Question. What do you think that should be the first step in the calculation of such indefinite integral $(1)$? I am not asking about the full deduction of such integral, only the first step (or the first and second ones) to calculate the indefinite integral. That is what is the idea to start to compute such indefinite integral. Many thanks.

Answer 1

Famously, $\operatorname{gd}y=2\arctan\exp y -\pi/2$. So your integral is $\int (2\arctan x -\pi/2)\arcsin' x dx$. By parts, you just need to obtain $\int\frac{\arcsin x dx}{1+x^2}$. Wolfram Alpha makes that look messy.