Integral of $(z^2 + x^2)^{-\frac{3}{2}}$ I am studying Griffiths Introduction to Electrodynamics, in which the following integral appears:
$$\int_{-L}^L\left({z^2+x^2}\right)^{-\frac{3}{2}}\,dx$$
where $z$ denotes a constant, and $z\in \mathbb{R}$
The integration is done without much fuzz about it, as if it was simple.
I tried throwing substitution, and partial integration at it, which didn't bring me far.
I now stumbled upon the following video https://www.youtube.com/watch?v=KR22vdvL_3g which deals with a similar integral.
There, use is made of hyperbolic functions and some rather advanced looking identities.
The speaker also mentions that this integral was particularly nasty to solve.
I find it hard to believe that one of the first problems in a rather introductory text should be so difficult, especially because the integration is done in one step, without any explanation.
Are there simpler methods of doing this integral, ones that i am missing?
If not, i guess my question becomes: How to solve this integral?

Assuming that $z$ is a (real) constant. Observe that the function $f(x)=\frac{1}{\sqrt{(x^2+z^2)^3}}$ is an even function. So
\begin{align*}
I & = \int_{-L}^L \frac{1}{\sqrt{(x^2+z^2)^3}} \, dx\\
&=  2\int_{0}^L \frac{1}{\sqrt{(x^2+z^2)^3}} \, dx\\
&= \frac{2}{z^2}\int_{0}^{\arctan\frac{L}{z}} \cos u\, du && (\text{ let } x =z \tan u )
\end{align*}
Now this should be easy to solve.
