complex analysis - Using residue theorem to compute an integral

This most likely has a simple solution, but for some reason I keep getting a complex answer to this real integral. Here's the problem:

Let $S_{\epsilon} := \{x \in \mathbb{R} : |x-1| > \epsilon\}$. Compute

$$\lim_{\epsilon\rightarrow 0^{+}} \int_{S_{\epsilon}} \frac{x}{(x^{2}+4)(x-1)}dx$$
using the residue theorem.

Attempt: I think my best bet is to use the simple contour $\Gamma$ which is a semi-annulus about $z=1$ in the upper-half plane of outer radius $R$ and inner radius $\epsilon$. That is, $\Gamma = \gamma_{1} - \gamma_{2} + \gamma_{3} + \gamma_{4}$, where

$$\begin{align} &\gamma_{1}: t \in [1-R, 1-\epsilon],\\ &\gamma_{2}: \epsilon e^{it}, 0 \leq t \leq \pi, \\ &\gamma_{3}: t \in [1+\epsilon, 1 + R], \\ &\gamma_{4}: Re^{it}, 0 \leq t \leq \pi. \end{align}$$

Letting $f(z) := \frac{z}{(z^{2}+4)(z-1)}$, the residue theorem gives me $\int_{\Gamma} f = 2\pi i\text{res}_{2i}(f) = \frac{1}{2(2i-1)}$ ($2i$ is the only pole in int$(\Gamma)$ when $R$ is large). Hence I have the identity

$$\begin{align} \frac{1}{2(2i-1)} &= \int_{\gamma_{1}} f(z)dz - \int_{\gamma_{2}} f(z)dz + \int_{\gamma_{3}} f(z) dz + \int_{\gamma_{4}} f(z)dz \\ &\implies \frac{1}{2(2i-1)} + \int_{\gamma_{2}} f(z) dz = \int_{\gamma_{1}} f(z)dz + \int_{\gamma_{3}}f(z)dz + \int_{\gamma_{4}}f(z)dz. \end{align}$$

Letting $R \rightarrow \infty$ yields

$$\begin{align} \frac{1}{2(2i-1)} + \int_{0}^{\pi} \frac{i\epsilon^{2}e^{2it}}{(\epsilon^{2}e^{2it}+4)(\epsilon e^{it}-1)}dt = \int_{-\infty}^{1-\epsilon} f(t) dt + \int_{1+\epsilon}^{\infty} f(t)dt, \end{align}$$

where the integral over $\gamma_{4}$

$$\int_{0}^{\pi} \frac{R^{2}ie^{2it}}{(R^{2}e^{2it}+4)(Re^{it}-1)} dt$$

went to $0$ as $R \rightarrow \infty$ (needs justification...). I then want to let $\epsilon \rightarrow 0^{+}$ to get

$$\frac{1}{2(2i-1)} + 0 (\text{right?}) = \lim_{\epsilon \rightarrow 0^{+}} \int_{S_{\epsilon}} f(t)dt.$$

The problem with this is that the RHS should be real, while the LHS is not. Where did I go wrong?

Edit: I realized my parametrizations were off a bit. $\gamma_{2}$ should be $1 + \epsilon e^{it}$ and $\gamma_{4}$ should be $1 + Re^{it}$. Yet still, the $\epsilon$-integral still goes to zero it seems.

Answer

I made a silly mistake: the parametrization I used for $\gamma_{2}, \gamma_{4}$ was wrong! I should've shifted by $1$. In particular, the $\epsilon$-integral becomes $\pi i/5$. I also forgot to multiply the residue I calculated by $2\pi i$. This gives me a real number for an answer.