I want to prove that there exists a constant $c\ge 0$ such that $$ln\Big(\frac{e^{\frac{\theta}{ch(\theta)}}+\frac{1}{ch^2(\theta)}\Big(sh(\theta)ch(\theta)-1\Big)}{e^{-\frac{\theta}{ch(\theta)}}+\frac{1}{ch^2(\theta)}\Big(sh(\theta)ch(\theta)+1\Big)}\Big)\ge c\theta$$
for all $\theta>0$? This is what i tried to do:
The derivative of the left hand function $\frac{2sinh(\theta)cosh(\theta)}{ch^4(\theta)}(1-ch(2\theta))(\frac{2}{cosh^2(\theta)}-2cosh(\frac{\theta}{cosh(\theta)})\Big)+(cosh(\theta)+\theta sinh(\theta))(sinh(2\theta)sinh(\frac{\theta}{cosh(\theta)})+(cosh(\theta)+\theta sinh(\theta))(-2+\frac{2}{cosh^2(\theta)}cosh(\frac{\theta}{cosh(\theta)}))$