@Markiyan Hirnyk OK, I am not that much prepared to answer complex analysis questions. All I know is that when

1<u<phi

all the things under radicals are positive and there should be no problem for simplifying sqrt{u} and sqrt{u-1} in donominator. The result obtained by Mathematica justifies my claim. Am I wrong again? :)

@vv As you can see it is already assumed that

1<u<phi

and hence all the arguments under radicals will be positive! I am not complaning about using complex numbers. I am saying that why Maple is not responding correctly and nicely when we have assumptions! and I have not switched completely yet but I am just trying to compare them and decide which one does the job better. :)

@Markiyan Hirnyk Please take a look at the question again. I added the result of Mathematica too! As expected this has nothing to do with branch cuts as we are assuming that u is real. :)

@Axel Vogt I added a picture to the question which describes the result of Mathematica. Take a look at it. I think that this is a flaw in Maple programming! :)

@Markiyan Hirnyk Yeah, It seems to be a duplicate but  I am not satisfied with the answer there. 

Let me ask you this. If we know that

1<u<phi 

then how do you simplify the expression? I think that is valid to cancel those similar terms but why maple do not do this!? This has nothing to do with the branch cuts as we are talking about a real number u! Am I wrong? :)

Please pay attention to each of the cases above for simplification. In some of them maple has canceled sqrt{u-1} but it has not canceled sqrt{u} or vice versa. I think there should be problem or something that I am missing. :)

@Carl Love I do really enjoy talking with you. Also I appreciate the @vv's contributions in the discussions. Thank you all. :)

@vv I think that you made the things a little complicated. What I want is so clear!

I don't understand that why I cannot stay in the real number system?

I don't see any complications for this! Maple just need to understand the rules of simplification in this field and what is defined or not defined there. That's all.

@Carl Love The first sentence in the Maple help page about RealDomain is

By default, Maple performs computations under the assumption that the underlying number system is the complex field. The RealDomain package provides an environment in which computations are performed under the assumption that the basic underlying number system is the field of real numbers.

What do you get of this sentence? 

I just get that if you want to just work with Reals call this package. Of course, if you want to work with the field of Real numbers then you will not be pleased to use assume for every variable.

About the name, you are right! :) I wouldn't use that if the name was like that. However this does not cover the need for such package that ensures that you are in the Real number system.

@tomleslie As you mentioned it should be used with care since "symbolic" simplification does not consider mathematical issues. So I prefer to use some command that consider the mathematical things too. :) Thanks for your attention. :)

@Carl Love Thanks for the details. But sometimes we prefer to do the computations with Reals. I do appreciate your support and nice clarificaiton. :) I think its time for maple developer to fix these small in-compatibilities! :)

@Preben Alsholm Thanks for clarification. If this forum could support the LaTex then you could write your proof in a more beautiful shape! :)

@Preben Alsholm So why maple simplifies u/sqrt{u}? but does not simplify the other expression in my file?

@Carl Love Is there a way that I can report this so it can be fixed in future updates? It seems that even in Maple 2015 the problem exists!

@Carl Love Don't you know what's wrong here?

@Kitonum This is one of the reasons that sometimes I want to change my CAS! :D