From: Albert_Rich@msn.com   
      
   On Monday, September 28, 2015 at 11:18:59 AM UTC-10, Nasser M. Abbasi wrote:   
   > Hello Martin:   
   >   
   > Rubi 8.1 on Mathematica 10.1 returns partial result:   
   >   
   > Clear[x];   
   > Int[ArcCos[x^2 - Sqrt[1 - x^2]], x]   
   >   
   > x*ArcCos[x^2 - Sqrt[1 - x^2]] +   
   >   2*Int[x^2/Sqrt[1 - (x^2 - Sqrt[1 - x^2])^2], x] +   
   >     Int[x^2/(Sqrt[1 - x^2]*Sqrt[1 - (x^2 - Sqrt[1 - x^2])^2]), x]   
   >   
   > Mathematica 10.1 and Maple 15 return unevaluated.   
   >   
   > --Nasser   
      
   Thanks for inspiring the Rubi Rule of the Week!  The next release of Rubi will   
   include a new rule for integrating expressions of the form   
      
     arccos(a*x^2 + b*sqrt(c+d*x^2))   
      
   wrt x when b^2*c=1, and a very analogous one for arcsin.  They are currently   
   available for viewing at   
      
     http://www.apmaths.uwo.ca/~arich/RuleOfTheWeek.pdf   
      
   They use integration by parts followed by piecewise constant extraction to put   
   the integrand in a form that can be handled by Rubi's existing rules.  So   
      
    Int[ArcCos[x^2 - Sqrt[1-x^2]], x]   
      
   will return the close to optimal(?) antiderivative   
      
   x*ArcCos[x^2-Sqrt[1-x^2]] -   
     (2*x*(1-x^2+2*Sqrt[1-x^2]))/Sqrt[x^2*(1-x^2+2*Sqrt[1-x^2])] +   
     (x*Sqrt[-1+x^2-2*Sqrt[1-x^2]]*   
       ArcSin[1+Sqrt[1-x^2]])/Sqrt[x^2*(1-x^2+2*Sqrt[1-x^2])]   
      
   Albert   
      
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