On a certain integral representation for Hurwitz zeta functions A recent question On a certain integral representation for Dirichlet L-functions referenced an integral representation of $\zeta(s)$ due to Jensen that was new to me:
$$
(s-1)\zeta(s)=\frac{\pi}{2(s-1)}\int_{-\infty}^\infty\frac{(1/2+it)^{1-s}}{\cosh^2(\pi t)}\, dt.
$$
Looking for analogs for the Hurwitz zeta function, I found some interesting integral representations due to Hermite in Whittaker and Watson, based on Plana's Theorem.  (Also found in Wikipedia under the Abel-Plana formula).  Exercises there include still more representations of $\zeta(s)$ due to Jensen, published in L'Intermediaire des Mathematicians, 1895.  In that publication I found the result above; towards the proof Jensen only says "proven with the aid of Cauchy's Theorem."
Internet searching eventually led me to (23) on p. 92 of Series Associated with Zeta and Related Functions, by Srivastava and Choi, Kluwer, 2001: for $a>1/2$,
$$
\zeta(s,a)=\frac{\pi}{2(s-1)}\int_{-\infty}^\infty\frac{(a-1/2+it)^{1-s}}{\cosh^2(\pi t)}\, dt.
$$
No proof is given.  For $a=1$ this reduces to Jensen.
Can anyone supply details of the proof?

A side note, the journal L'Intermediaire des Mathematicians was new to me.  It seems to be a 19th century print analog of MathOverflow, consisting of questions posed by researchers, and answers to questions from previous issues.  Jensen was writing in reference to a question of Cesaro.
 A: The formula for the Hurwitz zeta function (actually valid for complex $a$ with $\operatorname{Re}(a) > \tfrac{1}{2}$) is proved in the recent preprint Computing Stieltjes constants using complex integration by Iaroslav Blagouchine and myself, in which we moreover obtain an analogous representation for generalized Stieltjes constants,
$$\gamma_n(a) = -\frac{\pi}{\,2(n+1)\,}\!\int_{-\infty}^{\infty} 
\!\frac{\,\log^{n+1}\!\big(a-1/2 + it\big)\,}{\cosh^2 (\pi t)}\,dt, \quad \operatorname{Re}(a) > \tfrac{1}{2}.$$
We were not aware that the version of the formula for the Hurwitz zeta function already existed in the literature (however, if no proof has been written down before, that part of the work was at least not wasted) -- thanks for the reference!
Edit: my coauthor points out that formula (23) in Srivastava and Choi actually is slightly different from the one you reproduced in the question which is the same as the one in our preprint. Of course, the principles to prove either formula should be similar.
A: All these are special cases of the Abel--Plana formula: under simple regularity
and growth condition which are easily given, for any $a$ with $\Re(a)>-1/2$ we have
$$\sum_{n\ge 0}f'(n+a)=-\dfrac{\pi}{2}\int_{-\infty}^\infty \dfrac{f(a-1/2+it)}{\cosh^2(\pi t)}\,dt$$
plus some correction terms due to the possible poles of $f'$ in the right
half-plane, if any.
