Mathematica : Will not perform integral

[Hepth]

I wish to integrate:
$$\int dE \frac{\sqrt{E^2-m^2}}{E} \log \left(\frac{E-\sqrt{E^2-m^2}}{E+\sqrt{E^2-m^2}} \right)$$

Integrate[ Sqrt[(El - m) (El + m)]/El Log[( (El -Sqrt[(El - ml) (El + ml)])/ (Sqrt[(El - ml) (El + ml)] + El))], El]

But it won't do it. Anyone have any ideas as to why? I've set the assumptions
$Assumptions = ml >= 0 && El >= ml

Is this just an impossible integral? It seems well behaved.

 

[Simon_Tyler]

Hi Hepth,

Just because an integral is well behaved, doesn't mean that it's integrable in a nice closed form. Luckily, for your integral, all Mathematica needs is a little hand holding.

By writing x = ml^2/El^2, you can make the integrand look like

Sqrt[1 - x] Log[(1 - Sqrt[1 - x])/(1 + Sqrt[1 - x])]

This is easily checked in Mma using your assumptions.
The integral of the above integrand is

2/3 (1-x+Log[1-Sqrt[1-x]]-(1-x)^(3/2) Log[(1-Sqrt[1-x])/(1+Sqrt[1-x])]+Log[1+Sqrt[1-x]])

I've attached the notebook to this thread,

Simon