From: fateman@cs.berkeley.edu   
      
   On 3/1/2014 5:28 AM, Jonas Matuzas wrote:   
   ....   
   regarding integrals in terms of elementary functions   (or well-known   
   constants ...)   
      
   >> RJF   
   >   
   > I agree with you. You can name my integral   
   > Matuzas[a]=Integrate[Cos[x]^4/(a^8 + x^8), {x, -\[Infinity],   
   > \[Infinity]}] as new function. But you have to do table of this   
   > integral properties: differentiation, integration, differential   
   > equation it supports, Fourier series, expressions with another   
   > functions.   
      
   It would still not be an elementary function, even if you had   
   all that information available.(Historical convention, I suppose.)   
      
      If you want to learn about   
   the search for a solution to the problem of (indefinite) integration   
   in terms of elementary functions, you can try to find work   
   by Liouville, and then a stream of work including Robert Risch,   
   but not ending there. Maybe Manuel Bronstein's work would be   
   most recent.   
      
     And everybody should use it :) How to evaluate numerical   
   > -it is no problem . Actually the same is with Ci, Si and ect...   
   > Numerically we know how to evaluate.  But the same is with elementary   
   > functions, is'n it?   
      
   There is a question about how to do definite integrals. Maxima   
   obviously used a residue method rather than substitution into   
   the indefinite integral.  Making a wrong choice in method can   
   waste a lot of time.   
      
   Maxima's residue methods are described in a PhD thesis by Paul S H. Wang   
   at MIT.   
      
      
   >   
      
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