<s>[INST] 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. 
 [/INST] 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 $(\approx 0.195262)$.
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