How to force Mathematica to "Together" a fraction under a Log and a Square Root

Question

Consider the following code:

Assuming[2 > u > 0 && 2 > x > 0, FullSimplify[Log[x]/Sqrt[4 - x^2] /. x -> 2 Sqrt[(2 - u)/(2 + u)]]]

The following output is returned:

(the TeX'd equivalent, screenshots for Mathematica output make me uneasy)

$$\frac{1}{8}\sqrt{2+\frac{4}{u}} \text{Log}\left[-4+\frac{16}{2+u} \right]$$

How can I force Together on the arguments?

i.e. I would like it to look something closer to this:

$$\frac{1}{8}\sqrt{\frac{4+2u}{u}} \text{Log}\left[\frac{8-4u}{2+u}\right]$$

FullSimplify, Together, Simplify, etc. don't change the output by any significant margin.

Answer 1

Try this:

expr = Log[x]/Sqrt[4 - x^2] /. x -> 2 Sqrt[(2 - u)/(2 + u)]

(* Log[2 Sqrt[(2 - u)/(2 + u)]]/Sqrt[4 - (4 (2 - u))/(2 + u)] *)

Then:

Simplify[expr // FunctionExpand, 2 > u > 0]

yielding this:

Have fun!