What is the indefinite integral $\int \sqrt{1-2\sqrt{x-x^2}} \ \mathrm dx$? What is
$$\int \sqrt{1-2\sqrt{x-x^2}} \ \mathrm dx$$
I have tried substituting everything and it doesn't seem to be working. Substituting trigonometry doesn't seem to work either. 

Since, once more, Jack D'Aurizio is much faster than myself, let us see what can be done with Taylor built at $x=0$ $$\sqrt{1-2\sqrt{x-x^2}}=1-\sqrt{x}-\frac{x}{2}-\frac{x^2}{8}-\frac{x^3}{16}-\frac{5 x^4}{128}-\frac{7
   x^5}{256}+O\left(x^{11/2}\right)$$ So, integrating $$\int_{0}^{a}\sqrt{1-2\sqrt{x-x^2}}\,dx =a-\frac{2
   a^{3/2}}{3}-\frac{a^2}{4}-\frac{a^3}{24}-\frac{a^4}{64}-\frac{a^5}{128}-\frac{7
   a^6}{1536}+O\left(a^{13/2}\right)$$ For $a=\frac 12$, this gives $\frac{14123}{32768}-\frac{1}{3 \sqrt{2}}\approx 0.195297$ which compares quite well to Jack D'Aurizio's value $(\approx 0.195262)$.
