I'm looking for a way to prove the equality of the following two integrals with $a > b > 0$. Thanks.
$$\int_0^{+\infty} \dfrac{1}{\sqrt{\left(t^2+\frac{(a+b)^2}{4} \right) \left(t^2+\frac{(a-b)^2}{4} \right)}} dt = 2 \int_0^{+\infty} \dfrac{1}{\sqrt{(t^2+a^2)(t^2+a^2-b^2)}} dt$$