Hints to solve an equation containing $\int\frac{dx}{x\sqrt{(1+x^{-4})^n-1}}$ for $n<0$

I have an integral equation I need to solve. One of the sides has an integral of the form:

$$\int\frac{dx}{x\sqrt{(1+x^{-4})^n-1}}$$

where, for my case, $$n < 0$$. Mathematica yields no solution for this (it just yields my input, it doesn't say that it has no solution), so I'm trying to rewrite it in any way I can to see if there is an analytical solution. So far, upon the substitution $$x'=1+x^{-4}$$, I've managed to write it as:

$$\int\frac{dx}{(x-1)\sqrt{x^n-1}}$$

with some other constant multiplicative factors that do not affect the solution. Again, for this, Mathematica yields no solution.

I'm looking for any help, hint or clue to solve this integral. Also, any identification of the structure of a special function is appreciated.

• Does this integral have limits in the problem you're working with? Commented Feb 15, 2023 at 1:09
• Yes. I am expecting to obtain solutions from 0 to an arbitrary value, say x'. I expect to obtain regular values at 0 for n<0. Commented Feb 15, 2023 at 1:13

If you try for particular cases, for $$n=-1$$ and $$n=-2$$, Mathematica provides simple expressions.
But, as soon as $$n\leq -3$$ appear nasty elliptic integrals of all kinds with complex arguments.