Does the integral $\int\limits_0^{\infty}\frac{\sin^{2}x}{x^{2}\ln(1+\sqrt x)} dx$ converge?

Question

Does the integral

$$\int\limits_0^\infty \frac{\sin^{2}x}{x^{2}\ln(1+\sqrt x)} dx$$

converge?

It's easy to check that $\int\limits_1^{\infty}\frac{\sin^{2}x}{x^{2}\ln(1+\sqrt x)} dx$ does converge, but I couldn't find the right method for either proving or disproving that $\int\limits_0^1 \frac{\sin^{2}x}{x^{2}\ln(1+\sqrt x)} dx$ converges.

Answer 1

We might check some equivalent for $x\mapsto \frac{\sin^2(x)}{x^2 \ln(1+\sqrt{x})}$ near zero.

hint: $$ \ln(1+x) = x + o(x)$$