$$\int \frac{1}{\cos ^3 x \sqrt{\sin 2 x}} d x=$$

If $$\int \frac{\sqrt{1-x^2}}{x^4} \mathrm{~d} x=\mathrm{A}(x)\left(\sqrt{1-x^2}\right)^{\mathrm{m}}+\mathrm{c}$$ for a suitable chosen integer $$\mathrm{m}$$ and a function $$\mathrm{A}(x)$$, where $$\mathrm{c}$$ is a constant of integration, then $$(\mathrm{A}(x))^{\mathrm{m}}$$ equals

$$\int\left(\frac{\tan \left(\frac{1}{x}\right)}{x}\right)^2 d x=$$

$$\int \frac{1}{(x+2)(1+x)^2} d x$$ has the value