**Resolution of the Final ODE**

We use the
__gfun__
package to solve for the solution
which corresponds to the integral to be computed. Due to its integral representation, this function is analytic at 0, hence admits a Taylor expansion at 0. We proceed to compute a closed form for
by summation of this expansion. To this end, we determine a recurrence equation on the coefficients of the Taylor expansion using
__gfun[diffeqtorec]__
:

`> `

*Maple*
readily solves this recurrence:

`> `
**rsol:=rsolve(ore,u(n));**

After rearranging the terms in the sum, it is obvious that is non-zero for even only.

`> `
**collect(map(normal,rsol,expanded),u,factor);**

We perform the corresponding change of variable, ,

`> `
**subs({n=2*p,(-1)^n=1,RootOf(_Z^2+1)^n=(-1)^p},");**

so that
*Maple*
can sum the Taylor series:

`> `
**sum("*a^(2*p),p=0..infinity);**

`> `
**h:=collect(value(expand(")),u);**

It only remains to evaluate and . We first compute and find it is 0 by inversion of limits. Let

`> `

be the integrand. We have:

`> `

In the same way, each coefficient of the Taylor series for the integral is obtained by inversion of limits. In particular,
, but
*Maple*
is not capable of integrating:

`> `
**kappa=int(coeff(series(normal(diff(f,a,a)),a=0),a,0)/2,x=0..infinity);**

(This integral for
cannot be computed by a call to
*int*
using the Release 4, but the next release will probably be able to integrate it.)

We obtain the following form for :

`> `
**-combine(normal(-subs({u(0)=0,u(2)=kappa},h)),ln,symbolic);**

It only remains to be proved that . We do not do it, since computing this last integral which is a constant lies outside the scope of the theory of holonomy. With this example, we have reduced the problem of evaluating a parametrized integral to the evaluation of a non-parametrized integral. In case there were no closed form for , we could at least perform a simple numerical evaluation and return a result in terms of this numerical value and the series above for .