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