[Calculus] Having trouble parsing some steps in using u-substitution in an integral

The steps are as follows:

(Here's an image in case you don't use the LaTeX extensions or scripts)

Take the integral:

[; \int{ \cfrac{1}{{({R}^2 + x^2)}^{3/2}}\, \mathrm{d}x ;]

For the integrand [; \cfrac{1}{{({R}^2 + {x}^2)}^{3/2}} ;], (assuming all variables are positive) substitute:

[; {x} = {R}\tan ({u}) ;] and [; {d}x = {R}\sec^2{(u)} {d}u ;].

Then:

[; {{({R}^2 + {x}^2)}^{3/2}} = ({R}^{2} \tan^{2}{(u)} + {R}^2)^{3/2} = {R}^3 \sec^{3}{(u)} ;]

and [; {u} = \tan^{-1}{(\cfrac{x}{R})} ;]:

Results in:

[; = {R}\int{\cfrac{\cos{(u)}}{{R}^{3}}} {d}u ;]

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I understand these up until this step:

I don't understand where that inverse tangent comes from and how that suddenly works into making the result simplify into what it does

... [; {R}^3 \sec^{3}{(u)} ;] and [; {u} = \tan^{-1}{(\cfrac{x}{R})} ;]:

[; = {R}\int{\cfrac{\cos{(u)}}{{R}^{3}}} {d}u ;]

I know it's skipping some steps that may seem obvious to someone else, but if someone could go over it in detail it would help me with my understanding tremendously.

Thank you!