I'm going through David Tong's online lecture notes on String theory. At the end of section 1.1.2, where he introduces the einbein action
$$S=\frac{1}{2} \int d\tau (e^{-1}\dot{X}^2-em^2),\tag{1.8}$$
he claims that it retains reparametrization invariance through:
$$\tau \rightarrow \tilde{\tau}-\eta(\tau) \ \ , \ \ \delta e = \frac{d}{d \tau}(\eta(\tau)e) \ \ , \ \ \delta X^{\mu}=\frac{dX^{\mu}}{d\tau}\eta(\tau).\tag{1.10}$$
Can anyone help on how to prove invariance under these transformations? This is what I've worked out:
$$\tilde{S}=\frac{1}{2} \int d\tilde{\tau} (\tilde{e}^{-1}\dot{\tilde{X}}^2-\tilde{e}m^2)$$ $$= \frac{1}{2} \int d\tau (1-\eta'(\tau)) \left(\frac{1}{e+\frac{d}{d\tau}(\eta(\tau)e)} \eta^{2}(\tau)\dot{X}^2-\left(e+\frac{d}{d\tau}(\eta(\tau)e)\right)m^2 \right).$$
I'm not sure how to get $S$ again. The denominator is annoying.
