Series expansion for an integral - involving incomplete ellipticintegral

Hi,


I have a little question about a series expansion, which may be very
basic.

I would like to expand the integral of ArgCosh[1 + x^2 + a] between 0
and y >0, at first order in the small parameter a.

Obviously a naive approach using first order taylor expansion in the
integral doesn't work as it leads to a logarithmic divergency near 0.

I tried using Mathematica but it gives me a form involving incomplete
ellipitic integral that I do not understand how to deal with :
I*(EllipticE[I ArcSinh[Sqrt[1/a] x], a/(2 + a)] -
EllipticF[I ArcSinh[Sqrt[1/a] x], a/(2 + a)])
(there is an i in the first argument of the function). It looks like
this as the form cst + cst * a near 0 but I don't know how to compute
the cst exactly.

Does anyone know if this is a good point to start (as these forms come
from the exact primitivation of ArgCosh[1+x^2+a]), or if there is a
simpler way to find the answer ?


Thank you very much in advance.

Victor