Solve the following integral using substitution only?

Question

Can you solve the following integral using only substitution? $$\int \dfrac{dx}{\left(\sqrt{x^2-4}\right)^3}$$ I saw a solution to this which began with $x=2\sec(u)$, but is there another way to solve this? Thanks!

Answer 1

The substitution $x=t+\frac{1}{t}$ will do the job.

Integration by parts can also be used to reduce to a perhaps more familiar integral that can be trivially solved by the substitution $u=2\cosh x$.

Answer 2

Another way may be arisen by setting $$x^2-4=t^2x^2$$

Answer 3

One more substitution can be hyperbolic substitution.
Put $x=2\cosh{t}.$ Then $x^2-4=4\sinh^2{t},\;\;\; dx=2\sinh{t}\ dt,$ and $$\int \frac{dx}{\sqrt{(x^2-4)^3}}=\frac{1}{4}\int \frac{\sinh{t}\ dt}{\sinh^3{t}}=\frac{1}{4}\int \frac{dt}{\sinh^2{t}}=-\frac{1}{4}\coth{t}+C.$$