Integral of a weighted function

Question

I am trying to compute the following integral. Unfortunately, I do not have Maple(or any other program handy). I got stuck calculating it by hand: $$ \left(\int_0^tw(s)+t^p\int_t^{\infty}s^{-p}w(s) ds\right)^{1/p}, \quad p\geq1 $$ where $$ w(x)=\frac{1}{x(1-\ln x)^2}, \quad \textit{if} \quad 0<x<1 $$ and $$ w(x)=\frac{x^{p-1}}{(1+\ln x)^2}, \quad \textit{if} \quad x\geq 1 $$

Answer 1

Hint:

$$\int \frac1{x\left(1 - \ln x\right)^2}\mathrm d x = \frac1{1 - \ln x} + C$$