From: nma@12000.org   
      
   On 1/6/2016 3:54 AM, oldk1331@gmail.com wrote:   
   >   
   > In my opinion, many integration results given by Mathematica are   
   > consisted of complicated non-elementary functions or general   
   > solutions with large expression of symbolic radicals, when   
   > there exist simpler elementary results. I don't think Mathematica   
   > solves those integrals.   
   >   
      
   Yes, I noticed this. Also Mathematica sometimes gives an   
   inverseFunction as result when there seems to be simpler   
   result, here is an example:   
      
   DSolve[-Sqrt[Abs[y[x]]] + y'[x]==0,y[x],x]   
      
   {{y[x] -> InverseFunction[   
               -((2*2^(3/4)*   
                Hypergeometric2F1[1/4, 3/4,   
                 7/4, (2*Abs[Im[#1]] + I*(-2 + Conjugate[#1] +   
                   #1))/(4*Abs[Im[#1]])]*(Abs[Im[#1]] +   
                   I*(1 - Re[#1]))^(1/4)*(1 + I*Abs[Im[#1]] - Re[#1])*(1 -   
   #1))/(3*   
                Abs[Im[#1]]^(1/4)*   
                   (Im[#1]^2 + (1 - Re[#1])^2))) - (2*2^(3/4)*   
              Hypergeometric2F1[1/4, 3/4, 7/4,   
                   (2*Abs[Im[#1]] +   
                  I*(2*Abs[#1]^2 - Conjugate[#1] - #1))/(4*Abs[Im[#1]])]*   
                   ((-I)*Abs[Im[#1]] + Im[#1]^2 - Re[#1] + Re[#1]^2   
   *((Abs[Im[#1]] +   
                   I*(-Im[#1]^2 + (1 - Re[#1])*Re[#1]))/(Im[#1]^2 + Re[#1]^2))^(1/   
                 4)*(1 - #1))/(3*   
              Abs[Im[#1]]^(1/4)*(Im[#1]^2 + (1 - Re[#1])^2)) & ][x + C[1]]}}   
      
   Maple:   
      
   dsolve(diff(y(x),x)-abs(y(x))^(1/2) = 0,y(x))   
     x-piecewise(y(x) <= 0, -2*sqrt(-y(x)), 0 < y(x), 2*sqrt(y(x)))+_C1 = 0   
      
   I think Mathematica anaswer here is a bit too complicated. May be   
   due to the how they implement Risch algorithm? I do not know.   
      
   > On the other hand, when FriCAS returns an integral unevaluated,   
   > it means that this integral has been proven to have no elementary   
   > result by Risch algorithm.  While when other systems return an   
   > integral unevaluated, it is possible that elementary answer   
   > exist but the system can't find it.   
   > Eg. Integrate[(5*x^9-5*x^6+9*x^5-5*x^4-x^2+1)/(Sqrt[x^5+x-1]*(   
   ^5-x^2+x-1)),x]   
   >   
      
   I wonder what does FriCAS give for the above ode?   
      
   --Nasser   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)