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Sadly, this falls into the realm of many functions that do not possess an algebraic anti-derivative. This doesn't mean that its value doesn't exist, only that it cannot be expressed in terms of things such as trig functions, polynomials, or any other standard function.

One way you can try to express this value if needed could be through the use of a Taylor polynomial which for the first few terms comes out to be a

x-x^3/12-x^5/480-(19 x^7)/40320-(559 x^9)/5806080-(2651 x^11)/116121600....

While it may not help to directly calculate, you can express the value of anti-derivatives like this using something called an Elliptic Integral. This specific anti-derivative can be represented as 2E(x/2 | 2) where E is called the elliptic integral of the second kind which can be expressed as

E(p | k) = Integral from 0 to p of √(1-k²sin²(t))dt

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Q: What is the antiderivative of the square root of cos x?
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