Answer
$$ - \frac{1}{3}\csc 3\varphi + C$$
Work Step by Step
$$\eqalign{
& \int {\csc 3\varphi } \cot 3\varphi d\varphi \cr
& {\text{use the formula for indefinite integrals of trigonometric functions}} \cr
& \int {\csc ax} \cot axdx = - \frac{1}{a}\csc ax + C \cr
& {\text{letting }}a = 3,{\text{ and }}x = \varphi ,{\text{ we have}} \cr
& = - \frac{1}{3}\csc 3\varphi + C \cr
& {\text{check by differentiation}} \cr
& {\text{ = }}\frac{d}{{d\varphi }}\left( { - \frac{1}{3}\csc 3\varphi + C} \right) \cr
& {\text{ = }}\frac{d}{{d\varphi }}\left( { - \frac{1}{3}\csc 3\varphi } \right) + \frac{d}{{d\varphi }}\left( C \right) \cr
& = - \frac{1}{3}\left( { - \csc 3\varphi \cot 3\varphi } \right)\left( 3 \right) + 0 \cr
& = \csc 3\varphi \cot 3\varphi \cr} $$
