@Mariusz Iwaniuk What is this method _d01ajc  precisely?

Also what does maxintervals=300000 mean?

Setting your epsilon doesn't it determine the number of intervals in [0,1] ??

@vv Yes but how do you know it must have done that? Does it check everytime for convergence but since there is a period of 1 where nothing changes it assumes convergence??

Did you just try it out with 20 ?

@vv How do you have this insight?

@vv If necessary I will rewrite the problem tomorrow or so using different names, though I dont think it will change the essence.

Is it not clear what I mean?

If so, can somebody explain to me what information I‘m missing?

@Preben Alsholm  "The freezing that goes on is just a Maple technicality, which doesn't go to the essence of what is really going on. "

I'm aware of that. I just don't see why it is done here in particular as it seems he is freezing some arbitrary constants such as

_csgn(-(2*I)*RootOf(_Z^2+1, index = 1))

@Carl Love Why does he deem it necessary to freeze these objects anyhow?

@Preben Alsholm When using infolevel[int]:=5 then there seems to be lots of freezing going on which is however hidden.

For example this list

{_th[1] = ln(t-(1/2)*RootOf(_Z^2+1, index = 1)), _th[2] = ln(t+(1/2)*RootOf(_Z^2+1, index = 1)), _th[3] = exp(-(1/2)*s*(-2*ln(2)+ln(t-(1/2)*RootOf(_Z^2+1, index = 1))+ln(t+(1/2)*RootOf(_Z^2+1, index = 1))-(1/2)*RootOf(_Z^2+1, index = 1)*Pi*`freeze/R13`*(-`freeze/R13`+`freeze/R9`)*(-`freeze/R13`+`freeze/R10`))), _th[4] = exp(-(1/2)*s*(ln(-2*RootOf(_Z^2+1, index = 1))+ln(t+(1/2)*RootOf(_Z^2+1, index = 1))-(1/2)*RootOf(_Z^2+1, index = 1)*Pi*`freeze/R12`*(-`freeze/R12`+`freeze/R8`)*(-`freeze/R12`+`freeze/R10`)-ln(2*RootOf(_Z^2+1, index = 1))-ln(t-(1/2)*RootOf(_Z^2+1, index = 1))+(1/2)*RootOf(_Z^2+1, index = 1)*Pi*`freeze/R14`*(-`freeze/R14`+`freeze/R11`)*(-`freeze/R14`+`freeze/R9`)))}

is not very enlightening. Is it possible to show what is behind the frozen objects here?

@Preben Alsholm Yes^^ It seems to be iterating Risch algorithm forever, that's why the result is so big!

@Preben Alsholm So does FTOC use only lookup tables for the indefinite integral or does it also have access to series expansions (as in meijerg I suppose?)

Or how is it evaluating the indefinite integral?

@Preben Alsholm

## ftoc and ftocms uses the Fundamental Theorem of Calculus.

Out of curiosity: How do these methods proceed?

I presume they first convert the integrand into a corresponding series expression then integrate termwise (indefinitely) and convert the series back to an elementary expression? Finally the fundamental theorem is used?

@Preben Alsholm " ## NOTE added: Today I cannot reproduce the wrong ftoc and ftocms results for f itself. They now  agree with the correct  meijerg result. "

When was this added? So now it works for you? What did you do apart from this trick CarlLove mentioned?

@Preben Alsholm Hey again. I just wanted to say that I tried with Maple17 as well, and it seems the bug has been around for a long time.

@Preben Alsholm 

So is it possible to say, what other similar expressions relying on these methods will return an error/wrong result?

Can it be fixed?

@Axel Vogt I think the problem is that it once evaluates the crucial expression to

sqrt(I*(omega0^2-t))

and the other time to

sqrt(I)*sqrt(omega0^2-t)

These are however not the same.