@Mariusz Iwaniuk 

I could not reproduce the Mathematica answer with maple, which is in term of arcsinh

@dharr 

but if we simplify ln(x+i n) = ln(x)+pi then complex part vanishes and return to integral without I*n , why maple by using couchy principlevalue has obtained different answer?

@dharr 

Thanks for reply. we can consider 0 , infinity as limit

@Preben Alsholm 

@C_R 

Thanks for the answers but it does not give a solution yet!!

forgot to say \epsilon is small! 

@acer 

Thanks so much, very nice, it agrees with Mathematica answer!

But the original answer as I mentioned above is! i dont know how it is derived maybe with another form of change of variables ?

it is very similar to Mathematica and your answer ! but how it is derived

Question.mw 

I revised the integral which suffered from some problems. This new integral is only depends on "x". "n" is an even number. I would like to solve this integral in two regimes:

1) for a fixed "n" like "n=2" and as a function of "x" in a indefinite integral

2) as a function of "n" in a definite integral for "0<x<2pi".

Hope it could be solved.

Thank you all @acer @vv  @Axel Vogt

@acer That was my fault not define "g" and "J". I editted the file and put it again here. Actually the integral should include only "Zeta". I think it is ready now, please check the Maple file, is it solvable?

Question.mw

@acer But the problem remains unsolved because that was just a typo to pout the question here. Any idea to solve it?

@acer Thank you for your point and your comment. Yes that was my mistake, in "W" we should have "ln(varepsilon)".

@Rouben Rostamian  Thank you for your answer. 

I made a mistake in writing the denominator!

corrected.mw

Thank you for your time

@tomleslie 

Absolutely right , i want to obtain  [ Omega, alpha, beta, k] in terms of the others.

@Rouben Rostamian  

 I want to calculate the integral of " L"  for some solution of the equations of motion over some period of time. your solution just try to obtain equation of motion. the solution must be set into the Lagrangian and then you must integral over some period of time. The final result must have a shape like "A".

@John Fredsted  I made a mistake. you can take the end of time "t=t_0" or other thing else you like. It is a constant and doesn't matter. I don't understand what do you mean about "locf". where I mentioned to this quantity?

@John Fredsted yes I used from this package and obtained the equation of motion but my problems is the next step: when I must use from these solution, put into "L" and integrate it from "t=0" to "t=L". I must obtain at last a solution like "A" regardless what is the constants "a, b, s, L" 

PS: z(t)[h] is a constant as well and not be important

@Rouben Rostamian  

No I just want to solve the Euler-Lagrange equation for "L" and find equation of motions and then extermise "L" by them. I can't reach to a solution like "A" and I don't know where the problem is. I edited the maple file:

exfun2.mw