# Can these two indefinite integrals be evaluated in closed form?

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.

• what do we know about alpha? Nov 24 '14 at 15:37
• No constraints, $\alpha$ can be any real number. Nov 24 '14 at 15:39
• i think this will help you projecteuclid.org/… Nov 24 '14 at 15:42
• I doubt reading Risch's original work is helpful to the OP. Nov 24 '14 at 15:44
• 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. Nov 24 '14 at 19:04