Re: Inconsistent behaviour of Integrate

• To: mathgroup at smc.vnet.net
• Subject: [mg112445] Re: Inconsistent behaviour of Integrate
• From: "Sjoerd C. de Vries" <sjoerd.c.devries at gmail.com>
• Date: Wed, 15 Sep 2010 20:03:18 -0400 (EDT)
• References: <i6nea1\$g3h\$1@smc.vnet.net>

```Same here on my system

In[4]:= \$Version

Out[4]= "7.0 for Microsoft Windows (32-bit) (February 18, 2009)"

Cheers -- Sjoerd

On Sep 14, 11:12 am, Andreas Maier <andi... at web.de> wrote:
> Hello,
>
> I'm using Mathematica 7.0.1.0 on Linux x86 (64bit). I have a notebook
> file, where I integrate the same integral twice:
>
> In[1]:= Integrate[Sqrt[(x - 1/2)^2 + (y - 1/2)^2], {x, 0, 1}, {y, 0,
> 1}]
> Out[1]= 1/6 (Sqrt[2] + ArcSinh[1])
>
> In[2]:= Integrate[Sqrt[(x - 1/2)^2 + (y - 1/2)^2], {x, 0, 1}, {y, 0,
> 1}]
> Out[2]= 1/24 (4 Sqrt[2] + Log[17 + 12 Sqrt[2]])
>
> As you can see from the output, integrating the same integral a second
> time gives a different result. If I integrate the same integral a
> third and a fourth time I always get the second result again. Only if
> I restart the mathematica kernel, I get the first result again.
> The results are equivalent, since
>
> Log[17 + 12 Sqrt[2]] = Log[(1 + Sqrt[2])^4] = 4* Log[(1 + Sqrt[2]) =
= 4* ArcSinh[1]
>
> but somehow Mathematica seems to be able to do this simplification
> only once. Is this inconsistent behaviour a bug? Is there a
> possibility to give mathematica a hint, so that he always find the
> first solution 1/6 (Sqrt[2] + ArcSinh[1]) to the integral?
> From
>
> In[3]:= Expand[(1 + Sqrt[2])^4]
> Out[3]= 17 + 12 Sqrt[2]
>
> In[4]:= Factor[%]
> Out[4]= 17 + 12 Sqrt[2]
>
> I also figured that Mathematica doesn't seem to be able to factorize
> an expression like 17 + 12 Sqrt[2] into (1 + Sqrt[2])^4. Is this a
> known problem? Or should I use a different command to find this
> factorization?
>
> Sincerely,
> Andreas Maier

```

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