Integral result differ from Wolfram|Alpha

[Bùi Gia Nghĩa]

Hi!

I have used Sage Cell Server to integrate the function (ln(x)^2 - 1) / (x * ln(x)). It should resulted in (ln(x)^2) / 2 - |ln(ln(x))| + C, as noted by my textbook and Wolfram|Alpha, but instead resulted in 1/2*log(x)^2 - 1/2*log(log(x)^2). 

(I do notice that SageMath use log(a) to denote natural logarithm, so that's not the question here).

Anyone knows why it happen? I think that this is a bug from some system SageMath use to calculate this, but I am new to SageMath so have zero knowledge about the system.

Here is the exact code I input:

var("x")
f = (log(x)**2 - 1) / (x * log(x))
integral(f, x)

Thanks in advance!

[John H Palmieri]

Isn't log(log(x)^2) = 2 * log(log(x))? Is this your concern, or is it the absolute value?

[Oscar Benjamin]

On Mon, 13 Nov 2023 at 21:32, Bùi Gia Nghĩa <nghia...@gmail.com> wrote:
>
> Hi!
> I have used Sage Cell Server to integrate the function (ln(x)^2 - 1) / (x * ln(x)). It should resulted in (ln(x)^2) / 2 - |ln(ln(x))| + C, as noted by my textbook and Wolfram|Alpha, but instead resulted in 1/2*log(x)^2 - 1/2*log(log(x)^2).

Your textbook assumes that you are only interested in real
antiderivatives and so describes the antiderivative with abs that is
not complex-differentiable (i.e. not valid for non-real values of x).
The result from Sage is valid for e.g. x = 1+i whereas your textbook
result would be incorrect in that case.

--
Oscar

[Bùi Gia Nghĩa]

Oh that's why! Thank you because that is exactly the problem!

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