From: Albert_Rich@msn.com   
      
   In his post above, Martin points out Rubi fails on many “unnatural”   
   integrands.  Or as I prefer to call them: “contrived” integration problems.   
      
   In order to keep this project at least theoretically finite in nature, Rubi   
   does not worry about indefinite integrals having relatively simple   
   antiderivatives but complicated integrands.  Such gotcha problems are easily   
   generated by differentiating    
   simple expressions to produce complicated integrands.   
      
   Rather, the goal for Rubi is to produce optimal antiderivatives for ALL   
   members of a fixed set of general forms.  For example, one such form is   
      
        P[x] (d+e x^n)^q (a+b x^n+c x^(2 n))^p   
      
   where P[x] is any polynomial in x and the exponents n, p, q can be integer,   
   fractional or symbolic.  This is a huge class of expressions requiring   
   hundreds of rules to integrate.   
      
   In short, I contend that a symbolic integrator should strive to get such   
   coherent sets of real-world integrands before worrying about contrived ones…   
      
   Martin listed 6 MIT integration problems of the “natural” type that Rubi   
   should be able to integrate.  The version of Rubi currently under development   
   is able to find optimal antiderivatives for   
      
   #20 – 1/(Sin[x]+Sec[x])   
   #265 – ArcSin[x]*ArcCos[x]   
   #307 – x^(-Log[x])   
   #317 – Sin[4*ArcTan[x]]   
      
   And non-optimal antiderivatives for   
      
   #180 – Cos[x]*Cosh[x] + Sin[x]*Sinh[x]   
   #257 – (Cos[x]-Sin[x]) / (2+Sin[x])   
      
   Not quite sure why he considers “ill-posed” the MIT problem   
      
   #250 – Sqrt[1–ArcCos[Sin[x]]^2]   
      
   How should it be posed?   
      
   Albert   
      
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