If $$\mathrm{f}(x)=\int \frac{x^2 \mathrm{~d} x}{\left(1+x^2\right)\left(1+\sqrt{1+x^2}\right)}$$ and $$\mathrm{f}(0)=0$$, then $$\mathrm{f}(1)$$ is

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

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

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