Problem
Find $$\int \frac{\arcsin \sqrt{x}}{(1+x)\sqrt{x}}{\rm d}x.$$
My Try
Considering making a substitute, let $\arcsin\sqrt{x}=t$, where $0<t\leq \dfrac{\pi}{2}.$ Then we obtain $$x=\sin^2 t,{\rm d}x=2\sin t \cos t{\rm d}t.$$Thus,$$\int \frac{\arcsin \sqrt{x}}{(1+x)\sqrt{x}}{\rm d}x=2\int \frac{t\cos t}{1+\sin^2 t}{\rm d}t.$$
But I'm stuck here. How to go on with this?