The following two integrals are given in (Almost) Impossible Integrals, Sums, and Series (see Sect. $\textbf{1.55}$, page $35$),

$$i) \int_0^{\pi/2} \cot (x) \log (\cos (x)) \log ^2(\sin (x)) \operatorname{Li}_3\left(-\tan ^2(x)\right) \textrm{d}x$$ $$=\frac{109}{128}\zeta(7)-\frac{23}{32}\zeta(3)\zeta(4)+\frac{1}{16}\zeta(2) \zeta(5);$$ $$ ii) \ \int_0^{\log(1+\sqrt{2})} \coth (x) \log (\sinh (x)) \log \left(2-\cosh ^2(x)\right)\text{Li}_2\left(\tanh ^2(x)\right) \textrm{d}x$$ $$=\frac{73}{128}\zeta(5)-\frac{17}{64}\zeta(2)\zeta(3),$$ and they are evaluated by linking them to harmonic series of weight $7$, which in the book are calculated elementarily, mainly by simple and clever series manipulations.

My curiosity: Do you think it is possible to circumvent (at least for one of them) the work with harmonic series and get a different kind of solution?



You must log in to answer this question.