Re: Please help overcome problems with integration
- To: mathgroup at smc.vnet.net
- Subject: [mg72256] Re: [mg72229] Please help overcome problems with integration
- From: Daniel Lichtblau <danl at wolfram.com>
- Date: Sat, 16 Dec 2006 05:18:28 -0500 (EST)
- References: <200612151206.HAA00084@smc.vnet.net>
aaronfude at gmail.com wrote: > Hi, > > This is similar to my previous posts, but I have worked hard to come up > with the simplest possible example which highlights what ought to be > considered in bug in Mathematica. Please comment and suggest a work > around. > > F = Log[(-1 + d x + e Sqrt[f + 2g x + h x^2])]; > st = Integrate[F, x]; > > specific = st /. { d -> -2, e -> 10, f -> 1.3, g -> -1, h -> 1 }; > (specific /. x -> 1 ) - (specific /. x -> 0) > NIntegrate[F /. { d -> -2, e -> 10, f -> 1.3, g -> -1, h -> 1 }, {x, > 0, 1}] > > The results are: > > -1.25713 - 4.44^-16*I > > and > > 1.6552 > > Thank you very much in advance! > > Aaron Fude Comment: It's not a bug that the antiderivative, evaluated at endpoints, does not give the definite integral. The antiderivative has functions with branch cuts. Workaround: Use the parameter values in a definite integral. ee = Log[-1+d*x+e*Sqrt[f+2*g*x+h*x^2]]; rul = {d->-2, e->10, f->13/10, g->-1, h->1}; In[8]:= NIntegrate[ee/.rul, {x,0,1}] Out[8]= 1.6552 InputForm[ii = Integrate[ee/.rul, {x,0,1}]] Out[11]//InputForm= (-4352 - 272*Sqrt[55]*ArcTan[1/Sqrt[55]] - 42*Sqrt[22*(57 - (5*I)*Sqrt[55])]* ArcTan[(103*((-11*I)*Sqrt[5] + 27*Sqrt[11]))/ (97*(27*Sqrt[5] - (5*I)*Sqrt[11]))] - (42*I)*Sqrt[22*(57 + (5*I)*Sqrt[55])]* ArcTanh[(103*(11*Sqrt[5] - (27*I)*Sqrt[11]))/ (97*(27*Sqrt[5] + (5*I)*Sqrt[11]))] + 74*Sqrt[570 + (50*I)*Sqrt[55]]* ArcTanh[(103*(11*Sqrt[5] - (27*I)*Sqrt[11]))/ (97*(27*Sqrt[5] + (5*I)*Sqrt[11]))] - 2312*Log[7] + (21*I)*Sqrt[22*(57 - (5*I)*Sqrt[55])]*Log[21] - (21*I)*Sqrt[22*(57 + (5*I)*Sqrt[55])]*Log[21] + 37*Sqrt[570 + (50*I)*Sqrt[55]]*Log[21] - (21*I)*Sqrt[22*(57 - (5*I)*Sqrt[55])]*Log[3136] + (21*I)*Sqrt[22*(57 + (5*I)*Sqrt[55])]*Log[3136] - 37*Sqrt[570 + (50*I)*Sqrt[55]]*Log[3136] + 4352*Log[-3 + Sqrt[30]] - (21*I)*Sqrt[22*(57 + (5*I)*Sqrt[55])]* Log[249*I + 5*Sqrt[55] + (16*I)*Sqrt[3*(57 - (5*I)*Sqrt[55])]] + 37*Sqrt[570 + (50*I)*Sqrt[55]]*Log[249*I + 5*Sqrt[55] + (16*I)*Sqrt[3*(57 - (5*I)*Sqrt[55])]] + 37*Sqrt[570 - (50*I)*Sqrt[55]]* ((2*I)*ArcTan[(103*((-11*I)*Sqrt[5] + 27*Sqrt[11]))/ (97*(27*Sqrt[5] - (5*I)*Sqrt[11]))] + Log[(3*(-249*I + 5*Sqrt[55] - (16*I)*Sqrt[3*(57 + (5*I)*Sqrt[55])]))/ 448]) + (21*I)*Sqrt[22*(57 - (5*I)*Sqrt[55])]* Log[-249*I + 5*Sqrt[55] - (16*I)*Sqrt[3*(57 + (5*I)*Sqrt[55])]])/4352 + (-1360*ArcSinh[Sqrt[10/3]] + 272*Sqrt[55]*ArcTan[17/Sqrt[55]] - (42*I)*Sqrt[22*(57 + (5*I)*Sqrt[55])]* ArcTanh[(7711847*Sqrt[5] + (30516153*I)*Sqrt[11] + (1005856*I)*Sqrt[143*(57 - (5*I)*Sqrt[55])])/(18879807*Sqrt[5] + (2574065*I)*Sqrt[11])] + 74*Sqrt[570 + (50*I)*Sqrt[55]]* ArcTanh[(7711847*Sqrt[5] + (30516153*I)*Sqrt[11] + (1005856*I)*Sqrt[143*(57 - (5*I)*Sqrt[55])])/(18879807*Sqrt[5] + (2574065*I)*Sqrt[11])] + (42*I)*Sqrt[22*(57 - (5*I)*Sqrt[55])]* ArcTanh[(7711847*Sqrt[5] - (30516153*I)*Sqrt[11] - (1005856*I)*Sqrt[143*(57 + (5*I)*Sqrt[55])])/(18879807*Sqrt[5] - (2574065*I)*Sqrt[11])] + 74*Sqrt[570 - (50*I)*Sqrt[55]]* ArcTanh[(7711847*Sqrt[5] - (30516153*I)*Sqrt[11] - (1005856*I)*Sqrt[143*(57 + (5*I)*Sqrt[55])])/(18879807*Sqrt[5] - (2574065*I)*Sqrt[11])] + 2312*Log[43] - (21*I)*Sqrt[22*(57 - (5*I)*Sqrt[55])]*Log[43] + (21*I)*Sqrt[22*(57 + (5*I)*Sqrt[55])]*Log[43] - 37*Sqrt[570 - (50*I)*Sqrt[55]]*Log[43] - 37*Sqrt[570 + (50*I)*Sqrt[55]]* Log[43] + (21*I)*Sqrt[22*(57 - (5*I)*Sqrt[55])]*Log[118336] - (21*I)*Sqrt[22*(57 + (5*I)*Sqrt[55])]*Log[118336] + 37*Sqrt[570 - (50*I)*Sqrt[55]]*Log[118336] + 37*Sqrt[570 + (50*I)*Sqrt[55]]*Log[118336] + (21*I)*Sqrt[22*(57 + (5*I)*Sqrt[55])]* Log[2477*I + 65*Sqrt[55] + (38*I)*Sqrt[741 - (65*I)*Sqrt[55]]] - 37*Sqrt[570 + (50*I)*Sqrt[55]]*Log[2477*I + 65*Sqrt[55] + (38*I)*Sqrt[741 - (65*I)*Sqrt[55]]] - (21*I)*Sqrt[22*(57 - (5*I)*Sqrt[55])]* Log[-2477*I + 65*Sqrt[55] - (38*I)*Sqrt[741 + (65*I)*Sqrt[55]]] - 37*Sqrt[570 - (50*I)*Sqrt[55]]*Log[-2477*I + 65*Sqrt[55] - (38*I)*Sqrt[741 + (65*I)*Sqrt[55]]])/4352 In[13]:= Chop[N[ii]] Out[13]= 1.6552 Daniel Lichtblau Wolfram Research
- References:
- Please help overcome problems with integration
- From: aaronfude@gmail.com
- Please help overcome problems with integration