@Markiyan Hirnyk 

Yes, the interesting question is what method/algorithm Mathematica is using to get this result. Sadly, it appears like the access to this information is mostly forbidden to ordinary users. On the other hand, the CondiionalExpression construct is a very interesting feature introduced recently into Mathematica.

@Mac Dude 

May be that you mean the mechanism described in ?inttrans,addtable . And certainly, as shown in the log above, there is a stage of integral transforms lookup. So, it sounds reasonable to expect that added integral transform entries were used in this stage. But there is no documentation that I am aware describing such an extension of the capabilities of the lookup integration method ( e.g. nothing in ?int,methods ), and I have not experimented yet in this area.   

Actually, a table lookup method is tried in these three cases and fails, while method FTOC is the one that succeeds (here and in most other cases), or gets caught trying to (frequently while computing limits):

infolevel[IntegrationTools]:=3:
int(1/x^(1/3)/(x^2+2*x*cos(Pi/7)+1), x=0..infinity);
Definite Integration:   Integrating expression on x=0..infinity
Definite Integration:   Using the integrators [distribution, piecewise, series, o, polynomial, ln, lookup, cook, ratpoly, elliptic, elliptictrig, meijergspecial, improper, asymptotic, ftoc, ftocms, meijerg, contour]
IntegralTransform LookUp Integrator:   Integrand did not match an integral transform.
LookUp Integrator:   unable to find the specified integral in the table
memory used=37.7MB, alloc=43.1MB, time=0.57
Definite Integration:   Method ftoc succeeded.
Definite Integration:   Finished sucessfully.
                       / -----                            \
                       |  \     /          ln(-_R)       \|
                    -3 |   )    |1/6 --------------------||
                       |  /     |       /  3        Pi  \||
                       | -----  |    _R |_R  + cos(----)|||
                       \_R = %1 \       \           7   ///
                                   6       3      Pi
                    %1 := RootOf(_Z  + 2 _Z  cos(----) + 1)
                                                  7

I do not know whether this is a problem of "sophistication" or completeness of the table. You can observe the long list of matches it tries by:

infolevel[patmatch]:=3:
int(1/x^(1/3)/(x^2+2*x*cos(Pi/7)+1), x=0..infinity);
TableLookup:   
	matching expression: 1/x^(1/3)/(x^2+2*x*cos(1/7*Pi)+1)
	    against pattern: `/ODER`([`/KOMUTA`(`/POS`(1,`*`),`/ODER`([[`/POS`(1,tan,1\
), `/BIND`(1,1,`/X1`::linear(x))], [`/POS`(1,`^`,2), `/BIND`(1,1,`/X1`::linear(\
[...]

A first, cursory look at What's New in Mathematica 9 pages has given me the impression that several interface features look quite similar to those implemented in Maple, recently or time ago. For instance Next-Computation Suggestions Bar looks like Context-Sensitive Menus ( ?worksheet,expressions,manipulatecsm ), or Smart Popups ( ?worksheet,expressions,clickablemath ). And Context-Sensitive Input Assistant, looks like Command Completion ( ?worksheet,expressions,completecommand ) plus Palettes ( ?worksheet,expressions,palettes ).

Also, the new Advanced Hybrid and Differential Algebraic Equations deserves a thorough comparison with Differential-Algebraic Equations in Maple ( ?MaplePortal,DAE ). Idem for Units wrt the Units Package ( ?Units ). Finally, Surd was introduced among the Enhanced Core Algorithms, whose Maple equivalent existed, probably, since the earliest times ( ?surd ).

Wish lists are nice collections. Browsing them after some years (e.g. Maple 12 , 2008 poll) produces to me some nostalgia of seing so many good requests being unattended. And it is not clear to me the influence of those wish lists on the features that were later implemented. So, I wonder what is the purpose of these wish lists.

@Carl Love 

This is an elegant statement. Actually, many exported procedures are not documented...

@Carl Love 

This is an elegant statement. Actually, many exported procedures are not documented...

@Carl Love 

As an option, a graphic interface for the exploration of the content of modules and the code of procedures in libraries is available by executing LibraryTools:-Browse() (since Maple 9). This Maplet is potentially useful, but sadly somewhat buggy.

@Carl Love 

As an option, a graphic interface for the exploration of the content of modules and the code of procedures in libraries is available by executing LibraryTools:-Browse() (since Maple 9). This Maplet is potentially useful, but sadly somewhat buggy.

@Joe Riel 

As I see it, the "Reply to This" link occurs for "Comments", while "Answers" show a "Add Comment" link. So, if you mean the latter, such workaround is not good. Certainly, such generated Comment will, in principle, follow its parent Answer. But this Answer may exchange positions wrt other Answers as they get voted. And in so moving, they bring their child Comments with them. So, if I write something in such a child Comment about an Answer that was earlier to the parent Answer, and above of both at the time of writing,  what I wrote may loose sense as the Answers' relative positions are exchanged.

And I say "in principle" as it all depends on the Comment remaining a Comment. But this is at the will of the "supereditor". I have done it for some time, trying that my posts were not voted, but "someone" converted them to "Answers"...

Actually, Bryon said explicitly that they think that sorting posts by votes is a good idea (that came up during internal brainstorming), when he presented the plan for primes 2:

Change style of forums to a question/answer format
This is something that wasn't raised as a suggestion, but it came up during internal brainstorming and we think it's a good idea. [...]  Answers can be voted on so that the best ones rise to the top, and everyone can also engage in discussions by commenting on a particular question or answer.  What results is a community-generated library of questions and answers, much like we have now, but in a format that is more focused on promoting the best response.

Certainly, I think that this is an awful idea and its result is a mess. And I find that better than internal brainstorming, they should listen a bit outside.

It is almost two years and a half that users of this site have been requesting to keep the chronological order of the threads. I am convinced that there is no chance of a change because Bryon and Will have deaf ears to these requests. Implementing the current "populist" display mode is quite more complex that displaying the messages in their chronological order. Then, the only explanation that I can find for the implementation of this mode is their conviction that they are doing the "right" thing and they have given it top priority when designing primes 2. 

May be the consequence of a deleted spam post.

May be the consequence of a deleted spam post.

@PatrickT 

Actually, the automatic simplifier "lends" a bit of smartness to power as it processes some expressions that `^` constructs:

(-1)^(1/2); 
                                       I

(-1)^(-1/4);
                                        (3/4)
                                   -(-1)


(-1)^(7/6);
                                        (1/6)
                                   -(-1)

In this sense, `^` is an imperfect constructor as its output through the automatic simplification may be no longer of type `^`.