These involve separate cases $a=\tfrac13,\,a=\tfrac14,\,a=\tfrac16$ of, $${_2F_1\left(a ,a ;a +\tfrac12;-u\right)}=2^{a}\frac{\Gamma\big(a+\tfrac12\big)}{\sqrt\pi\,\Gamma(a)}\int_0^\infty\frac{dx}{(1+2u+\cosh x)^a}\tag1$$ The function $\eta(\tau)$ below is the Dedekind eta function. I. Case $a=\tfrac13$ Conjecture: There are an infinitely many algebraic numbers $\alpha, \beta$ such that $$H_1(\tau) =\,_2F_1\big(\tfrac13,\tfrac13;\tfrac56;-\alpha \big)=\beta$$ given by, $$\alpha = \frac1{4\sqrt{27}}\big(\lambda^3-\sqrt{27}\,\lambda^{-3}\big)^2$$ where $\lambda=\large{\frac{\eta(\frac{\tau+1}3)}{\eta(\tau)}}$ and $\tau=\frac{1+N\sqrt{-3}}2$ for any integer $N>1$. Examples: $$H_1\big(\tfrac{1+5\sqrt{-3}}2)=\,_2F_1\big(\tfrac13,\tfrac13;\tfrac56;-4 \big)=\tfrac3{5^{5/6}}$$ $$H_1\big(\tfrac{1+7\sqrt{-3}}2)=\,_2F_1\big(\tfrac13,\tfrac13;\tfrac56;-27 … Read more