From: fateman@cs.berkeley.edu   
      
   On 3/3/2014 7:24 AM, hebisch@math.uni.wroc.pl wrote:   
   > On Friday, February 28, 2014 11:33:56 PM UTC-5, Richard Fateman wrote:   
   >> I don't consider a solution that includes   
   >> Si, Ci, or hypergeometric functions as a solution   
   >> in closed form in terms of elementary functions.   
   >>   
   >> Unless there is no way of expressing the answer in   
   >> terms of elementary functions.   
   >>   
   >> After all, you could always decide that the   
   >> difficult integral in question deserves its own   
   >> name, say  FooI,  and then return the answer in terms   
   >> of FooI.   
   >   
   >   _Indefinite_ integral above can not be done using elementary   
   > functions.  For such integral 'li' and 'Ei' play the same   
   > as logarithms.  'Ci' and 'Si' are similar to 'atan'.   
   > As long as CAS can compute needed limits at infinity computing   
   > indefinite integral in terms of special functions   
   > is valid method of computing definite integral.   
   > And it is much more general than methods based on   
   > residue theorem.   
   >   
   > Waldek Hebisch   
   >   
      
   If your goal is not to compute a result in terms of elementary   
   functions but to allow terms of the "higher" or "special" functions   
   of applied mathematics, that's fine.   
      
     You just have to draw the   
   line somewhere, e.g. functions in Abramowitz and Stegun. or   
   the NIST digital library.   
      
   One of the uses of computer algebra systems is to find   
   explicit formulas when possible, and it is a puzzle whether   
   to use more functions, e.g. Si, Ci, Li;  or just express   
   those and other functions as hypergeometric functions.   
   So the idea of what is needed for an explicit solution   
   is somewhat fluid.  Macsyma for example, generally doesn't use   
   division internally.   a/b  is really a* (b^(-1)).   
   A trade-off between minimizing the number of different   
   functions and convenience.   For display purposes, Macsyma   
   prints a/b.  (free Maxima = Macsyma essentially)   
      
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