\left|\frac{\ln(1-\alpha^2x^2)}{x\sqrt{1-x^2}}\right |\le \left |\frac{\ln(1-x^2)}{x\sqrt{1-x^2}}\right |=\left |\frac{\ln(1+x)+\ln(1-x)}{x\sqrt{1-x^2}}\right |\le \frac{2|\ln(1-x)|}{x\sqrt{1-x^2}}\le K\frac{|\ln (1-x)|}{\sqrt{1-x}}