Integrals of the Form $\int\dfrac{dx}{a\sin^2x+b\cos^2x+c}$

To evaluate $\int\dfrac{dx}{a\sin^2x+b\cos^2x+c}$: (1) divide numerator and denominator by $\cos^2x$; (2) replace $\sec^2x$ by $1+\tan^2x$ in the denominator; (3) substitute $t=\tan x$, reducing the integral to the quadratic-denominator form $\int\dfrac{dt}{At^2+Bt+C}$ handled in 3.2.4; (4) evaluate using the appropriate special formula and re-express the answer in terms of $x$.

$\int\dfrac{dx}{3+2\sin^2x+5\cos^2x}$. Divide by $\cos^2x$: $\int\dfrac{\sec^2x}{3\sec^2x+2\tan^2x+5}\,dx$. Replace $3\sec^2x=3(1+\tan^2x)$: denominator becomes $5\tan^2x+8$. Put $t=\tan x$: $\frac15\int\dfrac{dt}{t^2+8/5}$, a Formula-1 shape: result $=\dfrac{1}{2\sqrt{10}}\tan^{-1}\dfrac{\sqrt5\tan x}{2\sqrt2}+c$. …