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I'm wondering whether any of these two indefinite integrals

$$\int \frac{1}{\sqrt{1+\alpha \sinh(x)^{-4/3}}}dx$$ $$\int \frac{\sinh(x)^{-4/3}}{\sqrt{1+\alpha\sinh(x)^{-4/3}}}dx$$

can be evaluated in closed form. I've tried a couple of substitutions, looked at integral tables, and asked Mathematica, to no avail so far.

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  • $\begingroup$ what do we know about alpha? $\endgroup$ Nov 24 '14 at 15:37
  • $\begingroup$ No constraints, $\alpha$ can be any real number. $\endgroup$
    – matimo2
    Nov 24 '14 at 15:39
  • $\begingroup$ i think this will help you projecteuclid.org/… $\endgroup$ Nov 24 '14 at 15:42
  • $\begingroup$ I doubt reading Risch's original work is helpful to the OP. $\endgroup$
    – GEdgar
    Nov 24 '14 at 15:44
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    $\begingroup$ Thanks. That brings them into slightly nicer looking form. For example, with further substitution $y=z^4$, the second integral becomes $\int \frac{z^2}{\sqrt{(z^6+1)(z^4+\alpha)}}dz$. However for Elliptic integrals I need a polynomial of degree at most 4 in the square root if I'm not mistaken. $\endgroup$
    – matimo2
    Nov 24 '14 at 19:04

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