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Difficult antiderivative


Using Mathematica (v5.1 to v.8) it is possible to obtain (in few
seconds) the following antiderivative

 

(1)  Integrate[ x ArcCosh[a/x]/Sqrt[r^2-x^2],x]

 

(a rather lengthy expression). 

 

It happens that I need 

 

(2)  Integrate[ ArcCosh[a/x]/Sqrt[r^2-x^2],x]

 

or equivalently

 

(3)  Integrate[ x^2  ArcCosh[a/x]/Sqrt[r^2-x^2],x]

 

 

I have tried a lot of tricks (by part integration etc..) but I was not
able to find it.

Q.: may be that the antiderivative does not exist?

The numerical integral  (b<r<a) 

 

NIntegrate[ ArcCosh[a/x]/Sqrt[r^2-x^2],{x,b,r}]

 

Don't give any problem.

Any help very appreciated (and considered).

Cheers, Rob

 

 

 

 

 

 

 

 


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