Albert Rich schrieb:   
   >   
   > 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...   
      
   ... to the extent that Timofeev covers the range of indefinite integrals   
   doable in elementary terms, Rubi appears to be pretty good already.   
   The evaluations for the integrals I mentioned were all marked as   
   unsatisfactory or sub-optimal in my files, and there are not many more   
   like them for Chapters 1,3,4,7,9 (I don't have comparable records for   
   Chapters 2,5,6,8).   
      
   >   
   > 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?   
      
   My brainwaves are all tied up at present. But how about implementing the   
   P_3^n recurrences from arXiv paper 1209.3758v2? The singly degenerate   
   recurrences would suffice for Examples 19 and 25 of Chapter 4 and   
   friends, whereas Example 99 needs the non-degenerate versions. Example   
   21, however, involves a self-degenerate P_1^m*Q_3^n; perhaps somebody   
   could work out the corresponding recurrences (a numerator A+B*x with two   
   coefficients would do)?   
      
   Ergo: New rules: yes, plethora: no.   
      
   >   
   > > 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).   
      
   Written this way, it does not seem to come out shorter. I have no strong   
   feeling here but my preference for continuity. An ASIN(SQRT(1-x^2)) in   
   the antiderivative also looks nice to me especially since it appears in   
   the integrand already.   
      
   Martin.   
      
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