Pizza-slice contour
I occasionally browsed a youtube video on "2023 MIT Integration Bee". I thought it is a competition between students taking the course of calculus. I took a try and it turns out that I was not able to even solve the first problem in the allowed 4 mins. The first problem is to solve \begin{equation} I \equiv \int_0^{\infty} \frac{\sqrt[3]{\tan x}}{(\sin x + \cos x)^2}dx\,.\tag{1}\end{equation} It is obvious that \begin{eqnarray}I &=& \int_0^{\infty} \frac{\sqrt[3]{\tan x}}{(\tan x + 1)^2} \frac{dx}{\cos^2 x} = \int_0^{\infty} \frac{\sqrt[3]{x}}{(x+1)^2}dx\\&=&-\left. \frac{\sqrt[3]{x}}{x+1}\right|_0^{\infty} + \int_0^{\infty} \frac{d\,\sqrt[3]{x}}{x+1}\\&=& \int_0^{\infty}\frac{dx}{x^3+1}\,.\tag{2}\end{eqnarray} I know how to solve the indefinite integral $\int\frac{dx}{x^3+1}$, but I was not able to accomplish the entire calculation within 4 mins. Here is another youtube video that provides step-by-step calculations. The result is \begin{equatio