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The post, Is there an integral for the golden ratio? gives numerous beautiful integrals for $\phi$. Some were just specializations of trigonometric evaluations such as, $$F(k)=\int_0^\infty \frac{x^{\pi/k-1}}{1+x^{2\pi}}dx =\frac{1}{2}\csc\Big(\frac{\pi}{2k}\Big)=\phi,\quad\text{at}\;k=5$$ However, integrals for phi's cousin the tribonacci constant $T$ seem to be harder to find. Phi appears in the pentagon, dodecahedron, etc, but $T$ also has a geometric context, the snub cube,

$\hskip2.8in$ Snub Cube

One integral I know for $T$ is, $$\beta\times \Bigl(\frac{T+1}{T}\Bigr)^2=\int_0^1 \frac{1}{\sqrt{(1-t^2)(1-k^2t^2)}}dt = 1.570983\dots$$ where, $$\beta=\frac{\Gamma\bigl(\tfrac{1}{11}\bigr)\, \Gamma\bigl(\tfrac{3}{11}\bigr)\, \Gamma\bigl(\tfrac{4}{11}\bigr)\, \Gamma\bigl(\tfrac{5}{11}\bigr)\, \Gamma\bigl(\tfrac{9}{11}\bigr)}{11^{1/4}(4\pi)^2}$$ $$k = \frac{1}{2}\sqrt{2-\sqrt{\frac{2T+15}{2T+1}}}$$ though it is a bit unsatisfying as $T$ appears in the integrand.

Q: Are there other nice integrals for the tribonacci constant $T$?

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A neat integral for the tribonacci constant $T$ involving the Dedekind eta function $\eta(\tau)$ is: $$\int_0^{\infty} \eta( i \, x)\,\eta(i \,11 x) \,dx = \frac{ \ln T}{\sqrt{11}} $$ There is also $$\int_0^{\infty} \eta( i x)\,\eta(i\, 23 x) \,dx = \frac{ \ln \rho}{\sqrt{23}} $$ where $\rho$ is the plastic constant. Check my question here for more details, and more similar integrals.

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  • $\begingroup$ Beautiful! A little experimentation showed that $$\int_0^{\infty} \eta( i x)\,\eta(i\, 5 x) \,dx = \frac{ \ln \big(\phi+\sqrt{\phi}\big)}{\sqrt{5}} $$ with the golden ratio $\phi$. $\endgroup$ – Tito Piezas III Nov 27 '16 at 12:28
  • $\begingroup$ @TitoPiezasIII Yes, I have made a list of all the polynomials corresponding to these integrals (see my question for information): pastebin.com/y5GHTBEE $\endgroup$ – nospoon Nov 27 '16 at 13:08

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