Step 1: Separate the integral into individual terms. $$ \int \left( \frac{3 \sin x}{\cos x} + \frac{\pi}{x} + 7 \right) dx = \int \frac{3 \sin x}{\cos x} \, dx + \int \frac{\pi}{x} \, dx + \int 7 \, dx $$ Step 2: Integrate the first term. Recall that $\frac{\sin x}{\cos x} = \tan x$. $$ \int \frac{3 \sin x}{\cos x} \, dx = 3 \int \tan x \, dx $$ The integral of $\tan x$ is $-\ln|\cos x|$. $$ 3(-\ln|\cos x|) = -3 \ln|\cos x| $$ Step 3: Integrate the second term. $$ \int \frac{\pi}{x} \, dx = \pi \int \frac{1}{x} \, dx $$ The integral of $\frac{1}{x}$ is $\ln|x|$. $$ \pi \ln|x| $$ Step 4: Integrate the third term. $$ \int 7 \, dx = 7x $$ Step 5: Combine all integrated terms and add the constant of integration $C$. $$ -3 \ln|\cos x| + \pi \ln|x| + 7x + C $$ The final answer is $\boxed{-3 \ln|\cos x| + \pi \ln|x| + 7x + C}$.