From: fateman@gmail.com   
      
   Maxima 5.45.1  returns the (unevaluated) integral in about 0.00 seconds.   
   Using quadpack numerical integration,  quad_qag()  it is possible to integrate   
   this function between any two points, (essentially instantaneously) and   
   therefore   
   plot it, if you wish.   
   Since the integrand does not have any symbolic constants, there is no need to   
   have   
   a surface or higher-dimensional plot.    
      
    The usefulness of the symbolic result you quote, from   
   Mathematica, is somewhat difficult to grasp; perhaps plotting it would be   
   easier to comprehend?   
      
   If I just use Mathematica's Integrate[]  , it fails to find a closed form. It   
   returns, after 10.21 seconds   
   {10.2188, \[Integral](-2 + 2^(1/3) (x + Sqrt[4 + x^2])^(2/3))^(   
      1/3)/(x + Sqrt[4 + x^2])^(1/9) \[DifferentialD]x}   
      
   My system, version 13.0.0  does not have IntegrateAlgebraic[]   
   so that's it.   
      
   While doing symbolic indefinite integration is an interesting test for   
   computer algebra systems,   
   and it is perhaps useful for helping with calculus homework, it has not, for   
   the most part,   
    entered the mainstream of applied scientific computing.   
      
   RJF   
      
      
      
      
    On Sunday, March 5, 2023 at 7:14:42 PM UTC-8, Sam Blake wrote:   
   > On Thursday, February 23, 2023 at 11:57:47 AM UTC+10, Sam Blake wrote:    
   > > Greetings,    
   > >    
   > > I was hoping the Risch-Trager-Bronstein algorithm of FriCAS and/or AXIOM   
   would compute the following integral in a reasonable amount of time    
   > >    
   > > -> integrate((x+(4+x^2)^(1/2))^(-1/9)*(-2+2^(1/3)*(x+(4+x^2)   
   (1/2))^(2/3))^(1/3),x)    
   > >    
   > > Was it wishful thinking? My computer has been running FriCAS 1.3.6 for   
   over an hour. The answer to this integral does not require any logarithms, so   
   it should be computable with the algebraic extensions of Hermite reduction,   
   right?    
   > >    
   > > Sam   
   > I guess not. This does make me question the practicality of the   
   Trager/Bronstein algorithm. Here's the solution from IntegrateAlgebraic in   
   Mathematica:    
   >    
   > In[5117]:= IntegrateAlgebraic[(-2 + 2^(1/3) (x + Sqrt[4 + x^2]   
   ^(2/3))^(1/3)/(x + Sqrt[4 + x^2])^(1/9), x] // Timing    
   >    
   > Out[5117]= {0.241908, (x (-2 + 2^(1/3) (x + Sqrt[4 + x^2])^(2/3))^(1/3))/(x   
   + Sqrt[4 + x^2])^(1/9) + ((-2 + 2^(1/3) (x + Sqrt[4 + x^2])^(2/3))^(4/3)   
   (-(1/10) - (11 (x + Sqrt[4 + x^2])^(2/3))/(20 2^(2/3)) - (x + Sqrt[4 +   
   x^2])^(4/3)/(20 2^(1/3))))/(x +    
   Sqrt[4 + x^2])^(10/9)}    
   >    
   > Sam   
      
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