Re: Re: Deleting Integrate[] transformation rule (some

*To*: mathgroup at smc.vnet.net*Subject*: [mg87725] Re: [mg87680] Re: Deleting Integrate[] transformation rule (some*From*: Daniel Lichtblau <danl at wolfram.com>*Date*: Wed, 16 Apr 2008 05:03:22 -0400 (EDT)*References*: <ftq5ab$it$1@smc.vnet.net> <ftscv1$b81$1@smc.vnet.net> <ftv91s$7ur$1@smc.vnet.net> <200804150954.FAA25209@smc.vnet.net>

UHAP023 at alpha1.rhbnc.ac.uk wrote: > Dear All, > Some progress on my original query. I tried the following where > the formulae below are expressions which integrate to elliptic > integrals of the first kind and are from mathematical handbooks. > > Unprotect[Integrate]; > > Integrate[1/Sqrt[1 - m_*Sin[phi_]^2], phi_] := > HoldForm[Integrate[1/Sqrt[1 - m*Sin[phi]^2], phi]] > > Integrate[1/Sqrt[(1 - v_^2)*(1 - k_^2*v_^2)], v_] := > HoldForm[Integrate[1/Sqrt[(1 - v^2)*(1 - k^2*v^2)], v]] > > Protect[Integrate]; > > > Subsequent attempts to integrate (eg. [Integrate[1/Sqrt[1 - a*Sin[b]^2], > b]) do indeed leave the integral undone as required. However more > involved expressions such as; > > In[52]:= > Integrate[Sqrt[Rvt^2 + (4*R^4)/Rx^4], R] // InputForm > > Out[52]//InputForm= > (R*Sqrt[Rvt^2 + (4*R^4)/Rx^4])/3 - > (I/3*Sqrt[2]*Rvt^2*Sqrt[Rvt^2 + (4*R^4)/Rx^4]* > Sqrt[1 - (2*I*R^2)/(Rvt*Rx^2)]* > Sqrt[1 + (2*I*R^2)/(Rvt*Rx^2)]*Rx^4* > EllipticF[I*ArcSinh[Sqrt[2]*R*Sqrt[I/(Rvt*Rx^2)]], > -1])/(Sqrt[I/(Rvt*Rx^2)]*(4*R^4 + Rvt^2*Rx^4)) > > still produce the unwanted EllipticF[]. My question is, is this; (a) > because my above HoldForm[] argument expressions are failing to > pattern-match the intermediate expressions produced by Integrate[], which > it then uses to produce results containing calls to EllipticF[], (b) > because Mathematica knows more expressions than I entered above > from my handbook which it can integrate up to EllipticF[], or > (c) something else? > > Any ideas? > > Thanks > Tom. > > Ps. The Email address in the header is just a spam-trap. If you are not averse to using a global flag variable, you can do it as follows. Unprotect[Integrate]; globalIntegrateFlag = True; Integrate[args__] := Block[ {globalIntegrateFlag=False, res}, res = Integrate[args]; If [FreeQ[res,EllipticF], res, HoldForm[Integrate[args]]] ] /; globalIntegrateFlag===True Examples: In[7]:= InputForm[Integrate[1/Sqrt[1 - m*Sin[phi]^2], phi]] Out[7]//InputForm= HoldForm[Integrate[1/Sqrt[1 - m*Sin[phi]^2], phi]] In[8]:= Integrate[1/Sqrt[1 - phi^2], phi] Out[8]= ArcSin[phi] (Even if you are averse to using a global variable, you can still do it that way; you'll just be less comfortable with it.) Daniel Lichtblau Wolfram Research

**References**:**Re: Deleting Integrate[] transformation rule (some progress)***From:*UHAP023@alpha1.rhbnc.ac.uk