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Inconsistent behaviour of Integrate


I'm using Mathematica 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,
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,
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?

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

Andreas Maier

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