I want to solve this recursion:
$$T(n) = 5T(\frac{n}{5}) + \frac{n}{lg(n)}$$
My attempt and issue:
None of the cases for master theorem apply here. I tried using Akra-Bazzi method (https://en.wikipedia.org/wiki/Akra%E2%80%93Bazzi_method) with
$f(n) = \frac{n}{lg(n)}$
The derivative of $f(n)$ satisfies the condition for Akra-Bazzi: $$\frac{d}{dn} f(n) = \frac{1}{lg(n)} - \frac{1}{ln(2)*lg^2(n)} = O(n)$$ Also I found that $p=1$ to satisfy the method's condition that $\frac{a}{b^p} = 1$.
Now the solution is given by this formula:
$$T(n) = \theta(n^p*(1-\int_{1}^{n} \frac{f(x)}{x^{p+1}} dx)$$
So I calculated the integral: $$\int\frac{f(x)}{x^{p+1}} dx = lg(lg(x)) + C$$, but with one of the limits being $1$ it diverges!
The weird thing is that according to Wolfram Alpha calculator, $T(n) = \theta(nlg(lg(n)))$.
So why is it true if the integral diverges? What am I getting wrong here?