@Axel Vogt 

Hey Alex, thanks for your maple sheet.

It indeed clarified my questions.

You used the value of "b" in some steps in order to arrive to the final result. But nevertheless, it is impressive that the integral can be evaluated in seconds!

It would be nice (elegant) if we could get a general expression that is a function of "b" where numerical "b" is just used in the end of the calculations and not in the intermediate steps.

Anyways, thanks so much!

@Axel Vogt 

Could you please elaborate on the following points:

- What do you mean by task in " the task is linear in b"? Just a polynomial of degree 1?

- What do you mean by " I transformed the task to the cube [0 .. 1]^3..." ? 

- "plotting gives nice shapes in that range". You just plotted "ms" from 0 to 1 to obtain the nice shapes?

- " Finally I get: Sol := b -> 26.0856584458258 + .616281090328306e-7*I -(21.9402269803097 + .514632049063567e-7*I)*b"

Is that a result for general b? I mean, did you keep b as a variable in ms or you performed the calculations with b = 0.7 (which in this case would make the result only good for b =0.7)? 

I really appreciate if you could explain those points! Thanks!

PS: In the stuff I am studying, b varies from -1<=b<0 and 0<b<1. Therefore, ms is a function of "x" and "b"

@Axel Vogt 

You are correct. read " Int( f(x), x = 0 .. x ) as Int( f(z), z = 0 .. x ) "

About the remaining of your comment, you lost me. You changed the variables of the problem to see if the integrals would be calculated faster?

@Carl Love 

There are some benchmark results in the literature that I used to compare and test this model/integral.

A simpler case, like you suggested, is b =0. In that case, ms is only a combination of trigonometrics and hyperbolics functions. 

Thanks for your input!

Thanks! @Carl Love 

Your procedure worked like a charm, though I do not understand all the code lines. I am a beginner with Maple.

I am solving some other integrals utilizing the same type of functions so I may post them later in case your procedure does not work with them.

And yes, I recognize that it is a mathematical error to have the same variable in the limits of integration and on the integrand. Not only Maple does not care about it but also most researchers in the particular field that I am studying on.

The expression has to do with "integration until a certain point" over the domain on a continuous system (rigid body). 

Anyways, thanks for your help. It will save me days from waiting out results!