simplification of trigonometric expression

Hello Maxima users,

I am computing the area of the ellipse using the polar equation 
r=p/(1+e*cos(theta)),
with 0<e<1 (not the simplest way, but I am playing with polar coordinates)
For this, I compute integrate(1/(1+e*cos(theta))^2,theta).
By hand, I find
f(e,theta):=2/(1-e^2)^(3/2)*atan(sqrt((1-e)/(1+e))*tan(theta/2))-e/(1-e^2)*(sin(theta)/(1+e*cos(theta)));

Maxima finds:
assume(e>0,e<1)$
integrate(1/(1+e*cos(theta))^2,theta);
define('g(e,theta),%);

g(e,theta):=2*(atan(((2*e-2)*sin(theta))/(2*sqrt(1-e^2)*(cos(theta)+1)))/(sqrt(1-e^2)*(e^2-1))-(e*sin(theta))/((cos(theta)+1)*(((e^3-e^2-e+1)*sin(theta)^2)/(cos(theta)+1)^2-e^3-e^2+e+1)))

Now, although it's quite obvious by visual inspection or numerical 
samples, I would like to make
Maxima find that f=g. I was not able to do that with ratsimp, trisimp, ...

Is there some function I don't know that could do that ?

Jean-Claude Arbaut