TITLE: How to choose the integration method for integrals involving powers and quotients of trigonometric functions?
QUESTION [5 upvotes]: I need help on these three integrals. Any hints on which method to use are greatly appreciated.
$$1)\ \int \frac{1}{\cos^4 x}\tan^3 x\mathrm{d}x$$
$$2)\ \int \frac{1}{\sin 2x}(3\cos x + 7\sin x)\mathrm{d}x$$
$$3)\ \int \frac{1}{\sqrt{x}}\sin^3 (3\sqrt{x})\mathrm{d}x$$

REPLY [3 votes]: Just because it wasn't mentioned explicitly I think it is worth mentioning on number 1.  The integrand is $\tan^3x\sec^4x$ which is equal to 
$$ \tan^3x(1+\tan^2x)\sec^2x$$
This is easily integrated as $\sec^2x$ is the derivative of $\tan x$.
$$ \tan^3x\sec^2x+\tan^5x\sec^2x $$