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Can this be solved in terms of elementary functions? If not, what is the approach?

$\int \sec x \sqrt{\tan x} \,\mathrm{d}x$

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    $\begingroup$ Please see here: wolframalpha.com/input/?i=integrate+sec+x+(tan+x)%5E(0.5) $\endgroup$
    – user371838
    Jan 28 '17 at 7:52
  • $\begingroup$ Why care about this? $\endgroup$
    – jimjim
    Jan 28 '17 at 8:21
  • $\begingroup$ this integral leads to an elliptic function $\endgroup$ Jan 28 '17 at 8:22
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    $\begingroup$ No, it cannot be solved in terms of elementary functions. Solving it gives elliptic integrals of the first and second kind. $\endgroup$ Jan 28 '17 at 11:46
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Let $\sqrt{\tan x}=t$.

Hence, $\frac{dx}{\cos^2x}=2tdt$, which gives $dx=\frac{2tdt}{1+t^4}$.

Hence, we need to calculate the following $\int\frac{2t^2}{\sqrt{1+t^4}}dt$

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