I am currently working on a project where I stumbled upon the integral
$$ \int \frac{\sinh \left(\frac{R}{2}\right)}{(\coth R - 6R \coth\left(\frac{R}{2}\right) + 9)^{1/4}} \,dR $$
where $R$ is a real variable. Is there a good way to integrate this? So far I haven't found anything fruitful, what worries me most is the fractional power in the denominator. Thanks for sharing your insight!
EDIT: I have tried the change of variable $\frac{R}{2} = \ln y$ and reduced the above integral to $$ 2^{1/4}\int \left(\frac{(y^2 - 1)^5}{ y^{12} + 17y^{10} - 17 y^8 - 24 y^8(y^2 + 1) \ln y - y^6 }\right)^{1/4} \,dy $$
I have some hope that this might help. Is there a good way to deal with the fact that the integrand is a fractional power, and with the logarithm in the denominator? Thanks again for sharing your insight!