Basti05b@aol.com schrieb:   
   >   
   > [...] In Volume 1 of my book, we use Riccati theorems to handle   
   > classical equations of physics and engineering.   
   >   
   > [...] Applications are ranging from solving integrations to   
   > differential equations. For example, as noted in DNA of mathematics   
   > book, the following function:   
   >   
   > S1: (t+2)/(t*sqrt((t+1)*(2*t^2+b*(t+1))));   
   >   
   > Where b is a non-zero real number, has integration of all rational   
   > functions.   
   > However Maple, and Mathematica, expresses them with Elliptic   
   > functions.   
   >   
      
   A response should be in order here. Straightforward testing shows that   
   your integrand S1 is Goursat-pseudoelliptic and its antiderivative in   
   terms of elementary transcendental functions can be obtained by   
   substituting x^2 = (t + 1)/(2*t^2 + b*t + b). The result is simply:   
      
   - 1/SQRT(b)   
     *ATANH((t^2 + b*t + b)/(SQRT(b)*SQRT((t + 1)*(2*t^2 + b*t + b))))   
      
   For example, FriCAS manages to evaluate this in elementary terms. But   
   the DNA of this class of integrands has been sequenced in Goursat's 1887   
   paper "Note sur quelques intégrales pseudo-elliptiques" already.   
      
   On the other hand, Fricas returns "Error detected: impossible" for the   
   cube-root pseudoelliptic 1/(x*((x - 1)*(x^2 - 2*q*x + q))^(1/3)) which,   
   however, may well be beyond all current integrators. Still, as the Risch   
   approach to general cube-root algebraics could not be made to work   
   during the last 25 years, one may doubt that it is practical at all.   
      
   Martin.   
      
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