While I was working on this question by @Vladimir Reshetnikov, I've conjectured the following closed-forms.
$$
I_0(n)=\int_0^\infty \frac{1}{\left(\cosh x\right)^{1/n}} \, dx \stackrel{?}{=} \frac{\sqrt{\pi}}{2} \frac{\Gamma\left(\tfrac{1}{2n}\right)}{\Gamma\left(\tfrac{1}{2}+\tfrac{1}{2n}\right)},
$$
for all $n\geq1$ real numbers. In another form:
$$
{_2F_1}\left(\begin{array}c\tfrac{1}{2n},\tfrac1n\\1+\tfrac{1}{2n}\end{array}\middle|\,-1\right) \stackrel{?}{=} \frac{\sqrt{\pi}}{n\,2^{1+\frac1n}} \frac{\Gamma\left(\tfrac{1}{2n}\right)}{\Gamma\left(\tfrac{1}{2}+\tfrac{1}{2n}\right)}.
$$
Another conjectured closed-form is
$$
I_1(n)=\int_0^\infty \frac{1}{\left(1+\cosh x\right)^{1/n}} \, dx \stackrel{?}{=} \frac{\sqrt{\pi}}{2^{1/n}} \frac{\Gamma\left(\tfrac{1}{n}\right)}{\Gamma\left(\tfrac{1}{2}+\tfrac{1}{n}\right)},
$$
for all $n \geq 1$ real numbers. In another form:
$$
{_2F_1}\left(\begin{array}c\tfrac1n,\tfrac2n\\1+\tfrac{1}{n}\end{array}\middle|\,-1\right) \stackrel{?}{=} \frac{\sqrt{\pi}}{n\,2^{\frac2n}} \frac{\Gamma\left(\tfrac{1}{n}\right)}{\Gamma\left(\tfrac{1}{2}+\tfrac{1}{n}\right)}.
$$
Here $\cosh$ is the hyperbolic cosine function, $\Gamma$ is the gamma function, and ${_2F_1}$ is the hypergeometric function.
Questions.
- $1^{\text{st}}$ question. How could we prove the conjectured closed-form for $I_0$ and $I_1$.
- $2^{\text{nd}}$ question. How could we show the equivalent hypergeometric forms.
- $3^{\text{rd}}$ question. There is a closed-form of $I_a(n) = \int_0^\infty \frac{1}{\left(a+\cosh x\right)^{1/n}} \, dx$ for $a\geq0,n\geq1$ real numbers in term of Appell $F_1$ function. Could we get a closed-form just in term of gamma function?
Best Answer
For the first one, $$\begin{align} \int_0^{\infty} (\operatorname{sech}x)^{2s}dx \\ &= \int_0^{\infty} (\operatorname{sech}^2x)^{s-1}\operatorname{sech}^2x\, dx \\ &= \int_0^{\infty} (1-\tanh^2x)^{s-1}\,\mathrm{d}(\tanh x)\\ &= \int_0^1 (1-x^2)^{s-1} \mathrm{d}x\\ &= \frac12 \int_0^1 (1-x)^{s-1} x^{-\frac12} \mathrm{d}x\\ &= \frac12 B(s,\frac12)=\frac{\sqrt{\pi}}{2}\frac{\Gamma(s)}{\Gamma(\frac12+s)} \end{align}$$ and so your conjecture is correct. For the second integral, let $x=2t$: $$\begin{align} \int_0^{\infty} \frac{1}{(1+\cosh x)^s}\mathrm{d}x \\ &= 2\int_0^{\infty} \frac{1}{(1+\cosh 2t)^s}\mathrm{d}t \\ &= 2\int_0^{\infty} \frac{1}{(2\cosh^2(t))^s}\mathrm{d}t\\ &= 2^{1-s} \int_0^{\infty} (\operatorname{sech}t)^{2s}\mathrm{d}t\\ &= 2^{-s} B(s,\frac12)=\frac{\sqrt{\pi}}{2^s}\frac{\Gamma(s)}{\Gamma(\frac12+s)} \end{align}$$