integration - What is the fast way to evaluate the following integral: $int{frac{sqrt{x^2+1}}{x^4}mathrm{d}x}$?

I am trying to evaluate the following integral:

$$\int{\dfrac{\sqrt{x^2+1}}{x^4}\mathrm{d}x}$$

I tried the trigonometric substitution: $u = \tan(x)$. Generally, The whole integral needs two substitutions: $u = \tan(x)$ then $v = \sin(u)$. In order to get rid of trigonometric functions, one needs to know that:

$$\sin(\arctan(x))=\dfrac{x}{\sqrt{x^2+1}}$$

My question is: What is the fast substitution that leads to the answer without passing by the above steps?