## Integral of ln x arcsin x dx

This is practically as complicated as solvable integrals get.

$\int \ln x\arcsin x\,dx=x \ln x\arcsin x- \int x(\frac{\arcsin x}{x} +\frac{\ln x}{\sqrt{1-x^2}}) \,dx=x\ln x\arcsin x-x\arcsin x-\sqrt{1-x^2}-\int \frac{x\ln x}{\sqrt{1-x^2}} \,dx=x\ln x\arcsin x-x\arcsin x-\sqrt{1-x^2}+\ln x\sqrt{1-x^2}-\int \frac{\sqrt{1-x^2}}{x} \,dx=x\ln x\arcsin x-x\arcsin x-\sqrt{1-x^2}+\ln x\sqrt{1-x^2}+\ln(1+\sqrt{1-x^2})-\ln x-\sqrt{1-x^2}+C$

$=\boxed{x\ln x\arcsin x-x\arcsin x-2\sqrt{1-x^2}+\ln x\sqrt{1-x^2}+\ln(1+\sqrt{1-x^2})-\ln x+C}$