Finding a constant that does not depend on a parameter

Question

We consider a parameter $\nu>0$. We note $b(\xi,x)=\nu(\xi^2+x^2)$ for all $\xi,x\in\mathbb{R}$.

I want to prove that there exists a constant $c>0$ such that

$$\int_0^{+\infty}e^{\Big(tanh(\frac{t}{2})-\frac{t}{2}\Big)b^2}dt\le \frac{c}{\nu^{\frac{2}{3}}}$$ for all $\xi,x\in\mathbb{R}$.

Please help me to do so. Thanks

Answer 1

Since $$ \int_0^{+\infty}e^{\Big(\tanh(\frac{t}{2})-\frac{t}{2}\Big)b^2}\,dt\ge \int_0^{+\infty}e^{\Big(-\frac{t}{2}\Big)b^2}\,dt=\frac{2}{b^2} $$ there is no such constant $c$ independent of $\xi,x\in\mathbb{R}$.