What are the harmonic conjugates of the following rational function?

Question

List all the harmonic conjugates of the following rational function. The integration is almost impossible, Symbolab and Microsoft Math fails to arrive at the answer: \begin{equation} \mu(x,y) =\frac{x^2+x+y^2}{x^2+y^2} \end{equation}

Answer 1

Look at

$\mu(x, y) = \dfrac{x^2 + x + y^2}{x^2 + y^2} \tag 1$

in polar coordinates:

$\mu(x, y) = \dfrac{x^2 + x + y^2}{x^2 + y^2} = \dfrac{r^2 + r\cos \theta}{r^2} = 1 + \dfrac{\cos \theta}{r}; \tag 2$

with

$z = x + iy = r\cos \theta + ir\sin \theta =re^{i\theta} \tag 3$

we have

$z^{-1} = r^{-1}e^{-i\theta} = \dfrac{\cos \theta - i\sin \theta}{r} = \dfrac{\cos \theta}{r} - i\dfrac{\sin \theta}{r} \tag 4$

is holomorphic on $\Bbb C \setminus \{0\}$; it follows that, as long as we stay away from $r = 0$, where $\mu$ is not in any event defined, that the harmonic conjugate of $\cos \theta / r$ is $-\sin \theta / r$, up to an addidive real constant. In $x$-$y$ coordinates, we have

$-\dfrac{\sin \theta}{r} = -\dfrac{r\sin \theta}{r^2} = -\dfrac{y}{x^2 + y^2}. \tag 5$

Thus, every harmonic conjugate of $\mu(x, y)$ is of the form

$\alpha - \dfrac{y}{x^2 + y^2}, \tag 6$

$\alpha \in \Bbb R$ a constant, in agreement with the comments of Thomas Andrews.