I am trying to parameterize the following integral
$$F(\lambda)=\int_{C j} e^{\lambda g(z)} h(z) d z$$
where $g(z)=z-z^3/3$ and $h(z)=z^{\mu-1}$ with the curve
$$\gamma(t)= \begin{cases}-t-1+i \sqrt{3\left((t+1)^2-1\right)} & t \geq 0 \\ t-1-i \sqrt{3\left((t-1)^2-1\right)} & t<0\end{cases}$$
which is a parameterization of $y^2=3(x^2-1)$. 
However, when I take the derivative I get $$\gamma^{\prime}(t)= \begin{cases}-1+\frac{i \sqrt{3}(t+1)}{\sqrt{t(t+2)}} & t \geq 0 \\ 1-\frac{i \sqrt{3(t-1)}}{\sqrt{(t-2) t}} & t<0\end{cases}.$$
which blows up when the curve passes through the origin. This integral should be convergent over the specified curve. What am I doing wrong for this to occur?