Examples of pairs of difficult integrals I’m looking for pairs of difficult definite integrals that are linked algebraically on a certain field without known change of variable or integration by parts from one integral to the other.
Here a ‘difficult’ integral means an integral without a known closed form.
Let’s see some examples. The second one is from Ramanujan.

$$ \int_0^1 \frac{\psi(x+1)+\gamma}{x} \:{\mathrm{d}}x = \int_0^1 \frac{x \ln x}{(1-x) \ln (1-x)} \:{\mathrm{d}}x \tag1$$

where $\psi:= \Gamma'/\Gamma$ is the digamma function and $\gamma$ denotes the Euler constant.

$$
\int_0^\infty\frac{t^{x}}{\Gamma(x+1)}{\mathrm{d}}x=e^{t}-\int_0^\infty\frac{e^{-tx}}{x(\pi^2 +\ln^2 x)}{\mathrm{d}}x, \quad t>0. \tag2
$$

Could you give other examples?
Thanks.
Addendum. 
Usually, when you try to evaluate an integral in closed form, you make use of transformations to bring it to a new integral which is a well-known integral.  You end up with an algebraic relation between your integrals. Let’s say your integrals are ‘chromatic’ with respect to a field. When two integrals are ‘chromatic’, evaluating one gives the answer for the other.
Your current tools among your transformations are: change of variable and integration by parts. What if we suppress these common tools?  What is left then?
This post is supposed to obtain some explicit examples of ‘chromatic’ difficult integrals that, as far as we know, are not linked from a change of variable or an integration by parts.
 A: $$
\int_0^{\infty} \frac{ dx }{ \Gamma (x) } = e + \int_0^{\infty} \frac{e^{-x}}{\pi^2 + \ln^2 x} \,dx = e + \frac{1}{\pi} \int_{-\pi/2}^{\pi/2} e^{\tan x} e^{ -e^{\ \tan x} } \,dx
$$
According to Wikipedia, it is unknown whether this constant can be expressed in closed form.
A: What about these pairs?
\begin{align}
    \nonumber(\alpha-\sin \alpha)&\int_{-\infty}^\infty\frac{d x}{(e^x-x-\cos \alpha)^2+(\alpha-\sin \alpha)^2}+\\
    &(\pi-\alpha+\sin \alpha)\int_{-\infty}^\infty\frac{d x}{(e^x+x+\cos \alpha)^2+(\pi-\alpha+\sin \alpha)^2}=\pi,\quad 0<\alpha<\pi
\end{align}
\begin{align}
    \nonumber(2\pi-\alpha+\sin \alpha)&\int_{-\infty}^\infty\frac{d x}{(e^x-x-\cos \alpha)^2+(2\pi-\alpha+\sin \alpha)^2}=\\
    &(\pi-\alpha+\sin \alpha)\int_{-\infty}^\infty\frac{d x}{(e^x+x+\cos \alpha)^2+(\pi-\alpha+\sin \alpha)^2},\quad 0<\alpha<\pi
\end{align}
It seems they are not linked by a change of a variable or integration by parts, and as far as I know none of these integrals have a closed form.
