a strange integral

• To: mathgroup at smc.vnet.net
• Subject: [mg71443] a strange integral
• From: "dimitris" <dimmechan at yahoo.com>
• Date: Sat, 18 Nov 2006 04:41:13 -0500 (EST)

```\$VersionNumber
5.2

Consider the following function

f[x_] := Log[1 + 1/Sqrt[x]]*(Log[1 + x]/x^(3/2))

Plot[f[x], {x, 0, 10}, PlotPoints -> 200]
Series[f[x], {x, 0, 2}]
Limit[f[x], x -> 0, Direction -> -1]

Here is the numerical estimate

NIntegrate[Log[1 + 1/Sqrt[x]]*(Log[1 + x]/x^(3/2)), {x, 0, Infinity},
PrecisionGoal -> 20, WorkingPrecision -> 40]
3.6705983269534578050

Strangely the following command "kill" the Kernel after several
minutes...

Integrate[Log[1 + 1/Sqrt[x]]*(Log[1 + x]/x^(3/2)), {x, 0, Infinity}]

What is more strange is that previous integral can be evaluated by
Mathematica
version 4.0; I don't check it myself by I adopted this integral from
the Mathematica
Guidebook for Symbolics of M. Trott where the computations take place
in v. 4.0 .

On the contrary, for v. 5.2 Integrate must be helped a little and then
the integration becomes trivial

integrand = Simplify[f[x]*dx /. x -> y^2 /. dx -> D[y^2, y], y > 0]
(2*Log[1 + 1/y]*Log[1 + y^2])/y^2

Timing[Integrate[integrand, {y, 0, Infinity}]]
N[%[[2]], 20]
{2.4689999999999994*Second, 4*Catalan + (5*Pi^2)/12 + Pi*(-2 + Log[2])}
3.6705983269534578050

Best Regards to all
Dimitris

```

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