Albert Rich schrieb: > > On Thursday, November 21, 2013 9:14:00 AM UTC-10, clicl...@freenet.de wrote: > > >>> Looking forward to Rubi4.3forte, > >> > >> Although not formally announced, Rubi 4.3 is now available for > >> downloading at > >> > >> http://www.apmaths.uwo.ca/~arich/ > >> > > ... Soul clap [its] hands and sing, and louder sing ... > > A truly Byzantine response; however, as an aged man I can relate to > studying monuments of its own magnificence.
The line was meant on a sanguine rather than elegiac note. Could it be that the coast has already hove in sight?
> > >> In addition to Euler's transformation it includes numerous > >> improvements including the use of rectification to produce continuous > >> antiderivatives after integrating trig expressions using the > >> substitution u=tan(x) or u=tan(x/2). The algorithm is described in > >> D.J.Jeffrey's 1997 paper "Rectifying Transformations for the > >> Integration of Rational Trigonometric Functions" available at > >> > >> http://www.apmaths.uwo.ca/~djeffrey/Offprints/trig-rec.pdf > > > > I haven't taken a close look yet but hope this is not what you called a > > "messy algorithm  not appropriate for an elegant, rule-based system; > >  it requires the host CAS provide a strong limit package and good > > algebraic simplification." Such requirements would severely limit the > > portability of Rubi. > > Rectification is implemented in Rubi as an optional, modular extension > to the back substitution routine used after successful integration by > substitution. If not implemented, the system will still return valid, > if not continuous, antiderivatives. > > Unfortunately, the substitutions u=tan(x) and u=tan(x/2) inherently > result in discontinuous antiderivatives. The only alternative I can > see to post-substitution rectification to produce continuous > antiderivatives would be to not use tangent substitutions at all. > Such substitutions essentially transform the integration problem from > the trig function world to the algebraic function world. Thus for > Rubi to avoid use of tangent substitutions, all the algebraic function > integration rules (currently 1013 rules) would have to be replicated > in the trig function world. Not a pleasant prospect...
You might employ SINT(f,x,x) in place of INT(f,x) and SINT(f,u,ux) for SUBST(INT(f,u),u,ux); this would make it easy to undo SUBST prior to evaluations that need to be continuitized. Presumably, the bulk of the 1013 algebraic function integration rules would go through without other changes, and just a number of terminal evaluation rules would have to be duplicated if ux = TAN(x) or TAN(x/2). Users would continue to see just INT and SUBST, of course, with SUBST(f,x,x) suppressed.
PS: In my "the more special a function, the fewer its integration rules", the word special should have been in quotes; "special" functions like generalized hypergeometric and Meijer-G functions _generalize_ less "special" ones, which is why their integration rules must be fewer.