On the integral $-\int\log\left(\binom{1-x}{6}+\binom{x}{6}\right)dx$, and its definite integral over the unit interval

Question

While I was playing with Wolfram Alpha online calculator I wondered that I know how to calculate with the help of this tool and my knowledges the first cases for integers $n\geq 1$ of this type of integral $$-\int_0^1\log\left(\binom{1-x}{n}+\binom{x}{n}\right)dx$$ and the corresponding indefinite integrals, where $\binom{a}{b}=\frac{a!}{b!(b-a)!}$ with $a!=\Gamma(a+1)$.

Question. Is it possible to find $$-\int\log\left(\binom{1-x}{6}+\binom{x}{6}\right)dx$$ in terms of standard mathematical functions? And is it possible to get the closed-form of the corresponding real part of such integral over the unit interval, that is $$-\int_0^1\Re\log\left(\binom{1-x}{6}+\binom{x}{6}\right)dx\,?$$ Justify your words. Many thanks.

Answer 1

With CAS help like Maple 2018 it possible to get the closed-form of the integral:

$$\int_0^1 \Re(\ln (\binom{1-x}{6}+\binom{x}{6})) \, dx= \pi\,\sqrt {103-8\,\sqrt {151}}+\pi\,\sqrt {103+8\,\sqrt {151}}-2 \,\sqrt {103-8\,\sqrt {151}}\arctan \left( \sqrt {103-8\,\sqrt {151}} \right) -2\,\sqrt {103+8\,\sqrt {151}}\arctan \left( \sqrt {103+8\, \sqrt {151}} \right) -\ln \left( 5 \right) -6 $$

Code:

evala(Re(int(ln(convert(binomial(1-x, 6)+binomial(x, 6), elementary)), x = 0 .. 1)));

#Pi*sqrt(103-8*sqrt(151))+Pi*sqrt(103+8*sqrt(151))-2*sqrt(103- 8*sqrt(151))*arctan(sqrt(103- 
#8*sqrt(151)))-2*sqrt(103+8*sqrt(151))*arctan(sqrt(103+8*sqrt(151)))-ln(5)-6

Maple can find for $n>7$, if n value is getting bigger the solutions is very complicated(bigger).

Example: for n=8 solution is written on 20 pages!.