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Re: Which symbolic method for this Integral ?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg82781] Re: Which symbolic method for this Integral ?
  • From: samuel.blake at maths.monash.edu.au
  • Date: Wed, 31 Oct 2007 06:03:54 -0500 (EST)
  • References: <fg6r2l$dsl$1@smc.vnet.net>

On Oct 30, 7:47 pm, Thierry Mella <thierry.me... at skynet.be> wrote:
> Hi,
>
> I'd like to know how Mathematica (5 or 6) integrate
> symbolicaly this integral :
>
> Integrate[1/(1 + Sqrt[1 + x] + Sqrt[1 - x]), x]
>
> Which method does it use ?
>
> Thanks,
> Thierry

Greetings,

While I am not aware how Mathematica calculates this integral, the
following may be of some help to you.

In general for purely algebraic functions high end CAS will use the
algebraic case of the Risch algorithm(while I say the "Risch
algorithm", most of the theory was discovered by J. Davenport, B.
Trager and M. Bronstein). This involves some very difficult
mathematics and can be very costly, especially when a substitution can
eliminate the algebraic terms from the integrand. As this integral has
two algebraic function extensions (over Q(x)) I can assure you that
doing this integral by hand using Risch is not practical. So I'll give
it a go using the good old method of substitution.

It's late in Australia, so PLEASE check this!! I believe that using
the substitutions

u = 1 - x

then

t = Sqrt[u]

then

t = Sqrt[2] Sin[v]

This will result in a trig rational integral which can be converted to
a rational integral using the trick given in every calculus book. Then
the rational integral can be calculated using the partial fraction
method. I would not expect the answer to be as nice as that given by
Mathematica, mainly because integrating a rational function using the
partial fraction method does not produce an answer in the minimal
algebraic extension field, whereas the Hermite reduction followed by
Lazard-Rothstein-Trager-Rioboo algorithm will! Also, I assumed t =
Sqrt[u] inverts to u = t^2 which causes further complications....

Sam



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