If $$\mathrm{I}=\int \sin (\log (x)) \mathrm{d} x$$, then $$\mathrm{I}$$ is given by

$$\int \frac{\mathrm{e}^x(1+x)}{\cos ^2\left(\mathrm{e}^x \cdot x\right)} \mathrm{d} x=$$

If $$\int \frac{\mathrm{d} x}{x \sqrt{1-x^3}}=\mathrm{k} \log \left(\frac{\sqrt{1-x^3}-1}{\sqrt{1-x^3}+1}\right)+\mathrm{c}$$, (where $$\mathrm{c}$$ is a constant of integration), then value of $$\mathrm{k}$$ is

$$\int \frac{\log (\cot x)}{\sin 2 x} d x=$$