How to evaluate this integral $\int\frac{\arcsin{\sqrt{x}}-\arccos{\sqrt{x}}}{\arcsin{\sqrt{x}}+\arccos{\sqrt{x}}}\cdot dx$?

Question

The integral is $$\int\frac{\arcsin{\sqrt{x}}-\arccos{\sqrt{x}}}{\arcsin{\sqrt{x}}+\arccos{\sqrt{x}}}\cdot dx$$

What I did was to sum up the denominator to equal $\frac\pi2$ and then applied integration by parts on the remaining $\arccos{\sqrt{x}}$ and $\arcsin{\sqrt{x}}$ integrals but this process was very lengthy.

Can someone suggest a shorter way?

Answer 1

$\arcsin{\sqrt{x}}+\arccos{\sqrt{x}}=\frac{\pi}{2}$ and $$\int\arcsin{\sqrt{x}}=x\arcsin{\sqrt{x}}-\int\frac{x}{\sqrt{1-x}}\cdot\frac{1}{2\sqrt{x}}dx$$ Can you end it now?