Some integrals with problematic evaluation

[quinyu]

Hello, Mr. Parisse,

I have been told on the HP forum to show the screen shots, not the command line, thought it would work the same way here; my bad. I will post command line input/output here from here on.
As of the status... it is a mix. For some answers, they are just very complicated. For other answers, the included functions (floor or signum) are not analytic, so you cannot make, for example, a power series or Fourier series expansion from them. Whether or not this is a problem of course depends on your application. Nevertheless, I have shown alternatives that don't require these functions as such. I must confess I have mixed feeling about the absolute value, but I see its benefits in most cases, so I don't argue that one.
Furthermore, I find it bad when imaginary values are pulled into an expression when there is an alternative expression without them.
But there are cases when the output is just completely faulty. I believe I found 1/(a^5+x^5) as an example, where the integration is just partially executed. Don't get me wrong: I am not saying that the output contains mathematical errors - the error is that it leaves an integral term in, when the expression is fully integrable; in a manner, this is similar to what my problem is with the signum and floor functions: why use when not needed? Certainly, I know that integration in some cases becomes difficult; and therefore I did not include those cases where the ordinary hypergeometric function would be expected as part of the answer, seeing that the HP Prime doesn't use it, just like no other CAS-capable calculators on the market. It is however my hope that based on these examples, some improvements can be made to the integration algorithm, as to output a more compact or readable answer, and avoiding the unnecessary imaginary in some cases.

And thanks, Arno, I'll put the "assume" in place from now on, where it makes sense.