Are there any papers/references on variations of polygamma integrals? For example, while browsing on this site, I found the following integral (in one of the answers), where the answerer did not believe there was a nice closed form.
$$\int_{0}^{1} x^2 \psi(2-x^4) \mathrm dx$$
I am familiar with the following eight results from this site (and MathOverflow). I will continue to add more results that I stumble upon.
$$\int_{1}^{\infty}x \psi^{(n)}(x) \mathrm dx = (-1)^{n+1}\Big[(n-2)!\zeta(n-1) + (n-1)!\zeta(n)\Big]$$
$$\int_{0}^{\infty} x^{\alpha - 1} \Big[\psi(b+x) - \psi(a+x)\Big] \mathrm dx = \frac{\pi}{\sin(\alpha \pi)}\Big[\zeta(1-\alpha, a) - \zeta(1-\alpha, b))\Big]$$
$$\frac{1}{7\pi} \int_{0}^{\pi} x(\pi - x)\Big[\psi \big(\frac{\pi + x}{2 \pi}\big) - \psi \big( \frac{x}{2\pi}\big)\Big] \mathrm dx = \zeta(3)$$
$$\int_{0}^{\infty} \psi^{(2)} (1+x) \ln (x) \, dx = \frac{\pi^2}{6} \left( \gamma + \ln (2\pi)-12 \ln A +1\right)$$
$$\int\limits_0^1 B_n(x)\psi(x+1)dx = (-1)^{n-1} \left(n \,\zeta’(1-n)- B_n H_{n-1}\right) + \sum\limits_{k=1}^n {\binom n k} \frac{B_{n-k}}{k}$$
$$\int_0^1 B_{2k+1}(x)\: \psi (x+1) \:dx=(-1)^{k+1}\frac{(2k+1)!}{(2\pi)^{2k+1}}\pi \: \zeta(2k+1)-\sum_{j=0}^{2k}\!\frac{ {{2k+1}\choose j} B_j}{2k+1-j}$$
$$\frac1{2\pi i} \int\limits_{-i\infty}^{i\infty} \Gamma^4\left(\frac12 + z\right) \Gamma^4\left(\frac12 - z\right) \psi\left(\frac12 - z\right)\ \mathrm dz = -\frac{2\pi^2}3 - \frac{2\gamma \pi^2}{3} - \zeta(3) $$
$$ \int_{0}^{\infty} \ln x\left[\ln \left( \dfrac{x+1}{2} \right) - \dfrac{1}{x+1} - \psi \left( \dfrac{x+1}{2} \right) \right] \mathrm{d}x = \dfrac{\ln^2 2}{2}+\ln2\cdot\ln\pi-1 $$
$$ \int_{-\infty}^{\infty} \frac{\psi \left(a- ip \right)}{\cosh^{2}(\pi p)} \, \mathrm dp = \frac{1}{\pi} \left(2 \psi\left(a+ \frac{1}{2} \right)+ (2a-1) \psi^{(1)}\left(a+ \frac{1}{2} \right) - 2 \right).$$
$$-\frac{\pi}{4}\int_{-\infty}^\infty \frac{\psi\left(\tfrac12+ip\right)+\psi\left(\tfrac12-ip\right)+2\gamma}{\cosh^2\pi p}e^{iap} dp = \frac{e^{a/2} \left(a^2+4 \text{Li}_2\left(1-e^{-a}\right)\right)}{4 \left(e^a-1\right)}$$
The derivations (in order from top to bottom) are ($1$), ($2$), ($3$), ($4$), ($5$), ($6$), ($7$), ($8$), ($9$), ($10$)
Furthermore, I am aware of the identity due to V. Adamchik, Polygamma functions of negative order Proposition $3$, Eq. $(17)$:
Let $n\in\mathbb{Z}_{\ge0}$ and $\Re z>0$, then \begin{align}\int_0^z x^n\psi\left(x\right) dx=&(-1)^{n-1}\zeta'(-n)+\frac{(-1)^n}{n+1}B_{n+1}H_n+\\ & \sum_{k=0}^n(-1)^kz^{n-k}{n \choose k}\left[\zeta'(-k,z)-\frac{B_{k+1}(z)H_k}{k+1}\right],\end{align} where $B_n$ and $B_n(z)$ are Bernoulli numbers and polynomials, $H_n$ are harmonic numbers, and $\zeta(s,z)$ denotes Hurwitz zeta function.
I am also aware of the following paper: A. Apelblat, Some Integrals of Gamma, Polygamma and Volterra Functions, that gives the following results (and many more!).
\begin{align*} &\int_{0}^{\infty} \frac{\psi(\alpha + x)}{\Gamma(\alpha + x)} \mathrm dx = \frac{1}{\Gamma(\alpha + 1)} \\\\ & \int_{1}^{\infty} \frac{\psi^{(1)}(z) - \psi(z)^2}{\Gamma(z)} \mathrm dz = \gamma\\\\ & \int_{0}^{\infty} \frac{1}{\Gamma(\alpha + z + 1)} [\psi(\alpha + z + 1)^2 - \psi^{(-1)}(\alpha + z + 1)] \mathrm dz = \frac{\psi(\alpha + 1)}{\Gamma(\alpha + 1)}\\\\ &\lim_{\alpha \rightarrow -1^+} \int_{0}^{\infty} \frac{1}{\Gamma(\alpha + z + 1)} [\psi^{(2)}(\alpha + z+1) - \psi(\alpha + z+1)^2 - 3\psi^{(1)}(\alpha + z + 1)\psi(\alpha + z+1)] \mathrm dz = 2\gamma \qquad \qquad \end{align*}
I'm very interested in these integrals, as sometimes they have delightful closed forms. Does anyone have any other results/references for integrals of the form/similar to (I am open to a wide range of such integrals, even if they are distantly related -- maybe even a log-polygamma integral)
$$\int_{a}^{b} f(x) \psi^{(m)}(g(x)) \mathrm dx$$
Your comments and inputs are highly appreciated :)
