Neo-Hookean tube pressure under twist and inflation ($p=\mu \log \frac{b_{0}a}{a_{0}b}+\frac{\mu }{2}\left(\frac{b_0^2}{b^2}-\frac{a_0^2}{a^2}\right)+\frac{\mu {\alpha }^2}{2}(b_0^2-a_0^2)$)

Here $b^2=b_0^2+a^2-a_0^2$. Substitute the neo-Hookean solid energy into the twist and inflation of an incompressible tube work identity. Differentiating with respect to $a$ gives the integrand $\mu \rho [{\rho }^{-2}-{\rho }^2/({\rho }^2+c{)}^2+{\alpha }^2]$, with $c=a^2-a_0^2$. Integrating between the two reference radii proves the displayed pressure when the outer surface is traction free.