:

Calculus1 step-by-step cycle

Maple

Could the Student:-Calculus1:-Hint routine detect that its answer(s) will result in an endless cycle (even if only cycles of length 2)? Maybe it could check, and not make such hints?

Below is just one example. Sure, its hint is not great here, due to which this is not a fantastic example. But I'm still wondering about cycling prevention for Rule&Hint in general.

This is Maple 13.01.

```> restart:
> with(Student:-Calculus1):

> Int((x^2+a)^(1/2)/(x+b^(1/2)),x);
/   2     1/2
|  (x  + a)
|  ----------- dx
|        1/2
/    x + b

> Hint(%);
1/2
[change, u = x + b   , u]

> Rule[%](%%);
/   2     1/2        /         1/2      2     1/2
|  (x  + a)          |  (b - 2 b    u + u  + a)
|  ----------- dx =  |  -------------------------- du
|        1/2         |              u
/    x + b           /

> Hint(%);
1/2
[change, u1 = -b    + u, u1]

> Rule[%](%%);
/   2     1/2        /    2     1/2
|  (x  + a)          |  (u1  + a)
|  ----------- dx =  |  ------------ du1
|        1/2         |    1/2
/    x + b           /    b    + u1
```

Of course, I don't mean the situations when the original integral is merely a part of a later form; that is a useful result when the original can be isolated. I mean when the repeated process cycles back to the very same form as the original.

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