From: Albert_Rich@msn.com   
      
   On Monday, December 15, 2014 8:03:29 AM UTC-10, clicl...@freenet.de wrote:   
      
   >> Unfortunately, not using PCE will require an exponential increase in   
   >> the number of rules required.  For example, Rubi currently has rules   
   >> for integrands of the form (a+b x)^m (c+d x)^n, and transforms   
   >> integrands of the form ((a+b x)^m (c+d x)^n)^p to that simpler form   
   >> using PCE.  Now it will have to have explicit rules for such   
   >> integrands.  Sigh...   
   >   
   > ... let's hope that Timofeev did a fairly thorough job.   
      
   Note quite sure of the relevance of your comment to the concern I raised over   
   the need for a plethora of new rules...   
      
   One solution I have considered is generalizing the existing rules to handle   
   non-integer powers of products of powers of linears.  For example, the   
   existing rule   
      
   Int((a+b x)^m (c+d x)^n, x) -->   
       (a+b x)^(m+1) (c+d x)^n / (b (m+1)) -   
       d n / (b (m+1)) Int((a+b x)^(m+1) (c+d x)^(n-1), x)   
      
   needs to be generalized to   
      
   Int(((a+b x)^m (c+d x)^n)^p, x) --> ???   
      
   without use PCE.  Can anyone help me out with the RHS?   
      
   > The continuitized antiderivative in Chapter 9 can be made even slightly   
   > simpler:   
   >   
   > INT(ASIN(SQRT(1 - x^2))/SQRT(1 - x^2), x) =   
   > 1/2*ASIN(x)*(pi - ASIN(SQRT(x^2)))   
      
   I find products of arcsines and arcsines of radicals objectionable.  Thus I   
   propose   
      
       1/2*pi*arcsin(x) - 1/2*x/sqrt(x^2)*arcsin(x)^2   
      
   as the optimal antiderivative of arcsin(sqrt(1-x^2))/sqrt(1-x^2).   
      
   Albert   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)