@spradlig 

I have the impression that Maxima is more close to what you are looking for than Maple or Mathematica. E.g. for automatic simplification:

(%i2) 8^(1/3)
(%o2) 2
(%i3) 8^(2/3)
(%o3) 4
(%i4) (-1)^(2/3)
(%o4) 1

And for plots:

plot2d(x^(2/3),[x,-1,1]);

@Kitonum 

Confirmed, it yields an error in Maple 12.02 and earlier (Linux 32-bit):

> `or`(0 <= 1, 0 <= 21, 0 <= 11);
Error, invalid input: or expects 2 arguments, but received 3

@Kitonum 

Confirmed, it yields an error in Maple 12.02 and earlier (Linux 32-bit):

> `or`(0 <= 1, 0 <= 21, 0 <= 11);
Error, invalid input: or expects 2 arguments, but received 3

@spradlig 

However, e.g. (-1)^(2/3) is not interpreted by WolframAlpha (Mathematica) as the square of the real cube root of -1, see here. And it agrees with the choice made by the automatic simplification of Maple in the float domain:

(-1.)^(2/3);
                 -0.5000000001 + 0.8660254037 I

I am not familiar with the Mac and its keyboard shortcuts. Does this combination command-period work with the CLI?

@Axel Vogt 

Yes, more or less I know it. And that scheme sounds nice a priori. But, on the one hand, it seems to require a pair of functions (F, invF), meaning  that you have to know the properties of F. But there are many special functions, Hypergeometric, Legendre, etc, where you do not have a "known F" at hand. So, it seems to me that this scheme is limited in practice to elementary functions.

And on the other hand, this scheme chokes with Maple, where e.g. series expansions and numerical evaluation of functions implement (presumably) the principal value approach, with branch cuts stemming basically from the chosen branch cut of ln(z) on the negative real axis. Certainly, this choice is conventional. As conventional as clock time, without which modern society cannot work...

@Axel Vogt 

Yes, more or less I know it. And that scheme sounds nice a priori. But, on the one hand, it seems to require a pair of functions (F, invF), meaning  that you have to know the properties of F. But there are many special functions, Hypergeometric, Legendre, etc, where you do not have a "known F" at hand. So, it seems to me that this scheme is limited in practice to elementary functions.

And on the other hand, this scheme chokes with Maple, where e.g. series expansions and numerical evaluation of functions implement (presumably) the principal value approach, with branch cuts stemming basically from the chosen branch cut of ln(z) on the negative real axis. Certainly, this choice is conventional. As conventional as clock time, without which modern society cannot work...

@Markiyan Hirnyk 

Yes, but I think that Edgardo has been testing against the Kamke sample for many years already. Somewhere I may keep some old copy. 

@Alex Smith 

The difference is that the  processing  of the definite integration tries, among other ones, the method ftoc (that would use the quickly computed primitive function), but it fails. Then it gets stuck when trying to compute it by the method meijerg.

@Alex Smith 

The difference is that the  processing  of the definite integration tries, among other ones, the method ftoc (that would use the quickly computed primitive function), but it fails. Then it gets stuck when trying to compute it by the method meijerg.

This is the n-th (n=?) version of this comparative review. It would be also interesting to see the historical evolution of these figures.

@Axel Vogt 

I see, you type IP for blackboard bold P. But then you should have typed something like IC instead of C, isn't it?

And you prefer the Riemann surface description of multivalued functions. The "problem" is that Maple, as well as most CAS, implement the principal value approach. Computationally, each approach, has pro and cons. The paper R.M. Corless, D.J. Jeffrey, Editors' corner: The unwinding number. SIGSAM Bulletin 116 (1996) provides an interesting comparison.

@Axel Vogt 

I see, you type IP for blackboard bold P. But then you should have typed something like IC instead of C, isn't it?

And you prefer the Riemann surface description of multivalued functions. The "problem" is that Maple, as well as most CAS, implement the principal value approach. Computationally, each approach, has pro and cons. The paper R.M. Corless, D.J. Jeffrey, Editors' corner: The unwinding number. SIGSAM Bulletin 116 (1996) provides an interesting comparison.

@Markiyan Hirnyk 

Certainly, it has been working with the plus signs, and failing with the minus signs, for over twenty years. It seems to me more than enough time for finding a fix for such a simple issue (factor out  -1). It took me a couple of minutes to write a transformation rule that handles both cases and more. So, it looks more like a sign of low priority for a very long time.

Yet, the recent reimplementation of this facility as a module, its extension with named rules related to its usage by math popups in the Standard GUI, and the fix that you have shown above, gives some hope of a nearby important improvement.

@Markiyan Hirnyk 

Certainly, it has been working with the plus signs, and failing with the minus signs, for over twenty years. It seems to me more than enough time for finding a fix for such a simple issue (factor out  -1). It took me a couple of minutes to write a transformation rule that handles both cases and more. So, it looks more like a sign of low priority for a very long time.

Yet, the recent reimplementation of this facility as a module, its extension with named rules related to its usage by math popups in the Standard GUI, and the fix that you have shown above, gives some hope of a nearby important improvement.