Integrating $\int x \csc^2(\,\ln(x)\,)\,dx$

Question

What is this integral?

$$\int x \csc^2(\,\ln(x)\,)\,dx$$

I haven't found any solution yet. It's the same as this one

$$\int \frac{x}{\sin^2(\,\ln x\,)}\,dx$$

So, how can I understand this integral in order to solve it?

Answer 1

A start

Recall that $$\sin x=\frac{e^{ix}-e^{-ix}}{2i}$$ So of course $$\sin\log x=\frac{x^i-x^{-i}}{2i}$$ $$\sin^2\log x=-\frac{(x^i-x^{-i})^2}{4}$$ Hence $$I=\int x\csc^2\log x\,\mathrm dx=-4\int\frac{x}{(x^i-x^{-i})^2}\mathrm dx$$ $$I=-4\int\frac{x}{x^{2i}+x^{-2i}-2}\mathrm dx$$ Which may be a little easier...