       integration using PSQL algorithm

• To: mathgroup at smc.vnet.net
• Subject: [mg52332] integration using PSQL algorithm
• From: Arturas Acus <acus at itpa.lt>
• Date: Wed, 24 Nov 2004 02:32:03 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```Dear group,

I have some reasons to suspect that some integral may have closed form
in terms of powers of Pi: a+b*Pi+c*Pi^2+d*Pi^3+e*Pi^4 (a,b,c,d,e being
algebraic numbers, probably with a=d=0). Assuming I can calculate the
integral numerically to desired precision, is it possible to verify this
hypothesis. As far as I know, the PSLQ algorithm, which helps to find
algebraic relations between real numbers, can be useful here. The only
(Mathematica related) reference I know is
http://library.wolfram.com/infocenter/MathSource/4263/
, but the example here is too simple.

As for details, the integral in consideration is

integral = (-6*(-1 + a)*(4*Sqrt[7 + a] + Sqrt*(5 + a)*Pi))/((5 +
a)*(7 + a)^(3/2)) -
(24*(-1 + a)*(6 + a)*Log[6 + a - Sqrt[5 + a]*Sqrt[7 + a]])/((5 + a)*(7
+ a))^(3/2)

with a= Cos[4* phi], integrated from 0 to Pi/2 (or to 2 Pi, no matter)
The problems arises from second term.

Mathematica 5.0 can't do it symbolically. How about 5.1?

Sincerely,
--
Arturas Acus <acus at itpa.lt>

```

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