$\int \frac{(1-4 \sin^2 x) \cos x}{\cos (3x+2)} dx =$

$\text{If } \int x[\log (1+x)]^3 dx = \frac{(1+x)^2}{16}(f(x)) + (1+x)(g(x)), \text{ then } f(x) + g(x) = $

If $\int \frac{(x^2-1)}{(x+1)^2 \sqrt{x(x^2+x+1)}} dx = A \tan^{-1}\left(\sqrt{\frac{x^2+x+1}{x}}\right) + C$,where $C$ is a constant,then $A$ equals to

$\int \frac{dx}{(1+\sqrt{x}) \sqrt{x-x^2}} = $

$\int \sqrt{x^2-8 x+7} \, dx = $ . . . . . . $+ C$.

If $\int(\sin x )^{\frac{-11}{2}}(\cos x )^{\frac{-5}{2}} dx = -\frac{p_1}{q_1}(\cot x)^{\frac{9}{2}}-\frac{p_2}{q_2}(\cot x)^{\frac{5}{2}}-\frac{p_3}{q_3}(\cot x)^{\frac{1}{2}}+\frac{p_4}{q_4}(\cot x)^{\frac{-3}{2}}+C,$ where $p_i$ and $q_i$ are positive integers with $\operatorname{gcd}(p_i, q_i)=1$ for $i =1,2,3,4$ and $C$ is the constant of integration,then $\frac{15 p_1 p_2 p_3 p_4}{q_1 q_2 q_3 q_4}$ is equal to . . . . . . .