differentiation under integral sign How do you find:

$$\int_0^\pi\frac{dx}{(5+3\cos x)^3}$$
and
$$\int_0^\pi\frac{\sin^2x }{(5+3\cos x)^3}dx$$
by differentiating under the integral sign?
I know you can get the result by using $\tan\left(\frac{x}{2}\right)$ substitution but I need to solve it via Feynman's Trick. 
I am given the general result that:
$$\int_0^\pi\frac{dx}{a+b\cos x}= \frac{\pi}{\sqrt{a^2-b^2}}$$

Let $I(a,b)=\int_0^\pi \frac{1}{a+b\cos(x)}\,dx=\frac{\pi}{\sqrt{a^2-b^2}}$.
Then, we see that
$$\begin{align}
\frac{\partial^2I(a,b)}{\partial b^2}&=\int_0^\pi \frac{2\cos^2(x)}{(a+b\cos(x))^3},dx\\\\
&=\color{blue}{2\int_0^\pi\frac{1}{(a+b\cos(x))^3}\,dx}-2\int_0^\pi \frac{\sin^2(x)}{(a+b\cos(x))^3}\,dx\\\\
&=\color{blue}{\frac{\partial^2I(a,b)}{\partial a^2}}-2\int_0^\pi \frac{\sin^2(x)}{(a+b\cos(x))^3}\,dx\\\\
\end{align}$$
