The function for harmonic numbers is given by $$H_x = \int_0^1 \frac{1-t^x}{1-t} dt \tag 1$$
For rational $x$, the formula is given by $$H_{x} = \frac{1}{x} + 2\sum_{k=1}^{\left\lfloor \frac{q-1}{2} \right\rfloor} \cos\left( 2\pi xk\right) \ln\left(\sin\left( \frac{\pi k}{q} \right) \right) - \frac{\pi}{2}\cot\left(\pi x\right) - \ln(2q) \tag 2$$ where $x = \frac{p}{q}, 0 < p < q$.
Is there a similar exact closed form for $x = m + \sqrt{n}$, where $m, n \in \mathbb{Z^+}$? I understand that $$H_x = H_{x-1} + \frac{1}{x} \tag 3$$ so it is only necessary to consider $0 < x < 1$. Alternatively, I could also only consider $m = 0$.
One thing I tried is writing the rational approximations for $x$. For example, if $x = -1 + \sqrt{2}$, then let $$p_n = \frac{(1+\sqrt{2})^n-(1-\sqrt{2})^n}{2\sqrt{2}}$$
Then let $x_n = \frac{p_{n-1}}{p_{n}}$. Here $x_n$ is the $n$th rational approximation to $x$. Then it is true that $$H_x = \lim_{n \to \infty} H_{x_n}$$ where $H_{x_n}$ can be evaluated using $(2)$.
Fully plugging everything in, I get $$H_{-1+\sqrt{2}} = 1+\sqrt{2} - \frac{\pi}{2}\cot\left(\pi \sqrt{2}\right)+\lim_{n \to \infty}(2\sum_{k=1}^{\left\lfloor \frac{p_n-1}{2} \right\rfloor} \cos\left( 2\pi (-1+\sqrt{2})k\right) \ln\left(\sin\left( \frac{\pi k}{p_n} \right) \right) - \ln(2p_n))$$
Simplifying a bit, I get $$1+\sqrt{2} - \frac{\pi}{2}\cot\left(\pi \sqrt{2}\right)+\frac{1}{2}\ln\left(2\right)+\lim_{n \to \infty}\left(2\sum_{k=1}^{\left\lfloor \frac{p_n-1}{2} \right\rfloor} \cos\left( 2\pi (-1+\sqrt{2})k\right) \ln\left(\sin\left( \frac{\pi k}{p_n} \right) \right) - n\ln\left(1+\sqrt{2}\right)\right)$$
However, the problem with this is that there seems to be no recognizable way to fully simplify $\lim_{n \to \infty} H_{x_n}$.
Alternatively, I could assume $m = 0$ and write $$H_{\sqrt{n}} = \int_0^1 \frac{1-t^{\sqrt{n}}}{1-t} dt$$ but this approach seems just as hopeless.
My questions:
$1.$ If a closed form for $H_x$ exists, where $x = m + \sqrt{n}$, what is it?
$2.$ If a closed form does not exist, why is that so? Would $H_x$ be a transcendental number?