TITLE: Closed-form of $\int_0^1 B_n(x)\psi(x+1)\,dx$
QUESTION [16 upvotes]: Is there a closed-form of the following integral?
$$I_n = \int_0^1 B_n(x)\psi(x+1)\,dx,$$
where $B_n(x)$ are the Bernoulli polynomials and $\psi(x)$ is the digamma function.
The motivation of the problem was this question. In this answer achille hui showed that for all $n>0$
$$\int_0^1 B_n\left(\left\{ \frac1x \right\}\right) \frac{dx}{x}
= -\int_0^1 B_n(u) \psi(1+u) du,$$
where $\{x\}$ is the fractional part of $x$.
By looking at specific $n$ values we could observe interesting patterns.
$$\begin{align}
I_0 & =  0\\
I_1 & =  1-\frac 12 \ln(2\pi) \\
I_2 & =  -\frac 12 + 2 \ln A \\
I_3 & = \frac{1}{12} - \frac{3}{4\pi^2}\zeta(3) = \frac{1}{12} - \frac{1}{8}\frac{\zeta(3)}{\zeta(2)}\\
I_4 & = \frac{1}{45} - 4\zeta'(-3)\\
I_5 & = -\frac{13}{360} + \frac{15}{4\pi^4}\zeta(5) = -\frac{13}{360} + \frac{1}{24}\frac{\zeta(5)}{\zeta(4)}\\
I_6 & = -\frac{1}{252}-6\zeta'(-5)\\
I_7 & = \frac{47}{1260} - \frac{315}{8\pi^6}\zeta(7) = \frac{47}{1260} - \frac{1}{24}\frac{\zeta(7)}{\zeta(6)}\\
I_8 & = -\frac{8}{1575}-8\zeta'(-7)\\
I_9 & = -\frac{1703}{25200} -\frac{2835}{4\pi^8}\zeta(9) = -\frac{1703}{25200} -\frac{3}{40}\frac{\zeta(9)}{\zeta(8)}\\
I_{10} & = \frac{2461}{83160} - 10\zeta'(-9)\\
\dots
\end{align}$$
where $A$ is the Glaisher–Kinkelin constant, $\zeta(3)$ is the Apéry's constant and in general $\zeta$ is the Riemann zeta function and $\zeta'$ is its derivative.

REPLY [3 votes]: The complete answer for the question with $\,n\ge 1\,$ is the union of the answers of Start wearing purple and Olivier Oloa (the sign of the right side of the equation $(*)$ is wrong there), we get:  

$$\int\limits_0^1 B_n(x)\psi(x+1)dx = (-1)^{n-1} \left(n \,\zeta’(1-n)- B_n H_{n-1}\right) + \sum\limits_{k=1}^n {\binom n k} \frac{B_{n-k}}{k}$$  

A well-functioning trick is to invert $(1)$ of Start wearing purple’s answer by using: 
$$b_{m-1}=\sum\limits_{v=0}^{m-1} {\binom m v} a_v \enspace \Leftrightarrow \enspace a_m=\frac{1}{m+1}\sum\limits_{v=0}^m {\binom {m+1} {v+1}} B_{m-v}b_v $$  
Note 1:
$$\sum\limits_{k=1}^n {\binom n k} \frac{B_{n-k}}{k}=\int\limits_0^1\frac{B_n(x)-B_n(0)}{x}dx$$
Note 2:
$$n\zeta’(1-n)- B_n H_{n-1}=\frac{d}{dt}B_t(1)|_{t=n}-H_n B_n(1)$$
$\hspace{4cm}$ Informations about $\,B_t(x)\,$ can be found here .