Rule-based integration
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From: Vladimir Bondarenko on 9 Jun 2010 08:14 There is much to say about the Rubi. One tiny example. The VM machine says: Mathematica 7.0.1.0 returns Integrate[1/(Sqrt[2] + Sin[z] + Cos[z]), z] unevaluated (!). Rubi returns -(2/(1 + (-1 + Sqrt[2]) Tan[z/2])) which is a correct answer. Best wishes, Vladimir Bondarenko VM and GEMM architect Co-founder, CEO, Mathematical Director http://www.cybertester.com/ Cyber Tester, LLC http://maple.bug-list.org/ Maple Bugs Encyclopaedia http://www.CAS-testing.org/ CAS Testing ----------------------------------------------------- "We must understand that technologies like these are the way of the future." ----------------------------------------------------- On Jun 4, 6:18 am, Albert <Albert_R...(a)msn.com> wrote: > I would like to announce the launch of the website > > www.apmaths.uwo.ca/RuleBasedMathematics > > It is dedicated to dedicated to demonstrating the numerous advantages > of the rule-based approach to automating mathematics. In systems > implemented using this approach, rules are expressed as elegant > mathematical formulas, rather than embedded in conventional program > code. > > As proof-of-concept, I have implemented an efficient and robust Rule- > based Integrator, nicknamed Rubi. Not only can Rubi compute the > antiderivative for a broad class of integrands, but the results are > often significantly superior to those produced by the commercial > computer algebra systems. > > The 1500 or so integration rules Rubi uses are freely available on the > website in both human and machine readable form. Also available is a > test suite of over 9400 integration problems developed in conjunction > with the rules. After reviewing the homepage, I recommend clicking on > "Highlights of the Indefinite Integration Test Results" for an eye- > opening comparison of the rule-based integrator (Rubi) with > Mathematica's and Maple's built-in integrators. > > Also if you have access to Mathematica 6 or better, there is a link > near the bottom of the homepage to download Rubi so you can verify the > results for yourself. Rubi also provides the option to show the rules > required to integrate expressions, along with the intermediate > results. I think this show-step ability has great potential > pedagogical and research value. > > Currently the website is pretty Spartan in format, and limited to > indefinite integration. However with the help of the computer algebra > community, I hope it evolves into a true repository of mathematical > knowledge. > > Aloha from Hawaii, > Albert D. Rich
From: Albert on 10 Jun 2010 02:23 For most integrands of the form 1/(a+b*Sin[x]+c*Cos[x]) Mathematica can use the rule Integrate[1/(a+b*Sin[x]+c*Cos[x]), x] --> 2 ArcTan[(b+(a-c)*Tan[x/2])/Sqrt[a^2-b^2-c^2]] / Sqrt[a^2-b^2-c^2] Obviously this rule cannot be used when a^2-b^2-c^2 equals 0, which is probably the reason Mathematica 7 cannot integrate 1/(Sqrt[2]+Sin[x] +Cos[x]). But, interestingly Mathematica is able to integrate 1/ (5+4*Sin[x]+3*Cos[x])... When a^2-b^2-c^2 equal 0, Rubi uses the rule Integrate[1/(a+b*Sin[x]+c*Cos[x]), x] --> -2/(b+(a-c)*Tan[x/2]) which gives the simple answer Vladimir reports. Albert
From: David Bernier on 10 Jun 2010 03:31 Albert wrote: > > For most integrands of the form 1/(a+b*Sin[x]+c*Cos[x]) Mathematica > can use the rule > > Integrate[1/(a+b*Sin[x]+c*Cos[x]), x] --> > 2 ArcTan[(b+(a-c)*Tan[x/2])/Sqrt[a^2-b^2-c^2]] / Sqrt[a^2-b^2-c^2] > > Obviously this rule cannot be used when a^2-b^2-c^2 equals 0, which is > probably the reason Mathematica 7 cannot integrate 1/(Sqrt[2]+Sin[x] > +Cos[x]). But, interestingly Mathematica is able to integrate 1/ > (5+4*Sin[x]+3*Cos[x])... > > When a^2-b^2-c^2 equal 0, Rubi uses the rule > > Integrate[1/(a+b*Sin[x]+c*Cos[x]), x] --> -2/(b+(a-c)*Tan[x/2]) > > which gives the simple answer Vladimir reports. This is quite interesting. Suppose we add real coefficients h and k for sin(2x) and cos(2x) respectively. Then, in the Mathematica notation, I arrive at this: Integrate[1/(a+b*Sin[x]+c*Cos[x]+h*Sin[2x]+k*Cos[2x]), x] Of course, there could be quite a few special cases here. David Bernier
From: Peter Pein on 12 Jun 2010 18:17 Hi Vladimir, nice to read you again :-)) my solution involves "PetSie.nb" which can be found at http://dl.dropbox.com/u/3030567/Mathematica/PetSie.nb In[3]:= <<PetSie` In[6]:= Assuming[Pi>x>0,ChangeVar[Tan[z/2]==t,Integrate[1/(Sqrt[2]+Sin[z]+Cos[z]),{z,0,x}],t]] Out[6]= (2 (-1+Sqrt[2])Sin[x/2])/(Cos[x/2]+(-1+Sqrt[2]) Sin[x/2]) there is a difference between the usage of a cas and simply asking a cas. Greetings, Peter Am Wed, 9 Jun 2010 05:14:21 -0700 (PDT) schrieb Vladimir Bondarenko <vb(a)cybertester.com>: > There is much to say about the Rubi. > > One tiny example. > > The VM machine says: > > Mathematica 7.0.1.0 returns > > Integrate[1/(Sqrt[2] + Sin[z] + Cos[z]), z] > > unevaluated (!). > > Rubi returns > > -(2/(1 + (-1 + Sqrt[2]) Tan[z/2])) > > which is a correct answer. > > > Best wishes, > > Vladimir Bondarenko > > VM and GEMM architect > Co-founder, CEO, Mathematical Director > > http://www.cybertester.com/ Cyber Tester, LLC > http://maple.bug-list.org/ Maple Bugs Encyclopaedia > http://www.CAS-testing.org/ CAS Testing > > ----------------------------------------------------- > > "We must understand that technologies > like these are the way of the future." > > ----------------------------------------------------- > > On Jun 4, 6:18 am, Albert <Albert_R...(a)msn.com> wrote: > > I would like to announce the launch of the website > > > > www.apmaths.uwo.ca/RuleBasedMathematics > > > > It is dedicated to dedicated to demonstrating the numerous > > advantages of the rule-based approach to automating mathematics. > > In systems implemented using this approach, rules are expressed as > > elegant mathematical formulas, rather than embedded in conventional > > program code. > > > > As proof-of-concept, I have implemented an efficient and robust > > Rule- based Integrator, nicknamed Rubi. Not only can Rubi compute > > the antiderivative for a broad class of integrands, but the results > > are often significantly superior to those produced by the commercial > > computer algebra systems. > > > > The 1500 or so integration rules Rubi uses are freely available on > > the website in both human and machine readable form. Also > > available is a test suite of over 9400 integration problems > > developed in conjunction with the rules. After reviewing the > > homepage, I recommend clicking on "Highlights of the Indefinite > > Integration Test Results" for an eye- opening comparison of the > > rule-based integrator (Rubi) with Mathematica's and Maple's > > built-in integrators. > > > > Also if you have access to Mathematica 6 or better, there is a link > > near the bottom of the homepage to download Rubi so you can verify > > the results for yourself. Rubi also provides the option to show > > the rules required to integrate expressions, along with the > > intermediate results. I think this show-step ability has great > > potential pedagogical and research value. > > > > Currently the website is pretty Spartan in format, and limited to > > indefinite integration. However with the help of the computer > > algebra community, I hope it evolves into a true repository of > > mathematical knowledge. > > > > Aloha from Hawaii, > > Albert D. Rich >
From: Vladimir Bondarenko on 30 Jun 2010 00:45
Hello Albert, The Cyber Tester's team is interested in improvements along the direction you offer, rule-based approach to automating mathematics. As you know, we do not test software ourselves; instead, this work is done by our VM machine. It is important to learn how much we could help you and the community in reinforcing Rubi as the very first step of your project. To know this, it would be nice and fair to learn, before we start presenting the results by the VM machine, 1) how much time have you spent writing the Rubi? 2) how much time you and your beta testers spent for QA? 3) is there a list of known bugs in the Rubi? About the time, ANY estimation however approximate is good. Best wishes from algorithmic Simferopol, Vladimir On Jun 4, 6:18 am, Albert <Albert_R...(a)msn.com> wrote: > I would like to announce the launch of the website > > www.apmaths.uwo.ca/RuleBasedMathematics > > It is dedicated to dedicated to demonstrating the numerous advantages > of the rule-based approach to automating mathematics. In systems > implemented using this approach, rules are expressed as elegant > mathematical formulas, rather than embedded in conventional program > code. > > As proof-of-concept, I have implemented an efficient and robust Rule- > based Integrator, nicknamed Rubi. Not only can Rubi compute the > antiderivative for a broad class of integrands, but the results are > often significantly superior to those produced by the commercial > computer algebra systems. > > The 1500 or so integration rules Rubi uses are freely available on the > website in both human and machine readable form. Also available is a > test suite of over 9400 integration problems developed in conjunction > with the rules. After reviewing the homepage, I recommend clicking on > "Highlights of the Indefinite Integration Test Results" for an eye- > opening comparison of the rule-based integrator (Rubi) with > Mathematica's and Maple's built-in integrators. > > Also if you have access to Mathematica 6 or better, there is a link > near the bottom of the homepage to download Rubi so you can verify the > results for yourself. Rubi also provides the option to show the rules > required to integrate expressions, along with the intermediate > results. I think this show-step ability has great potential > pedagogical and research value. > > Currently the website is pretty Spartan in format, and limited to > indefinite integration. However with the help of the computer algebra > community, I hope it evolves into a true repository of mathematical > knowledge. > > Aloha from Hawaii, > Albert D. Rich |