Evaluate
\documentclass{article}
\usepackage{bigints}
\begin{document}
\[
\bigint_0^1 (4x^2+4^2x^{2^2}+4^3x^{2^3}+\cdots)\frac{\log^2(x)}{x(1+x)} \textrm{d}x
\]
\end{document}
You should be able to solve this