Integration of trigonometric and hyperbolic functions
`I=int(dx)/(cos^(4)x).`

Answer

To solve the integral \( I = \int \frac{dx}{\cos^4 x} \), we can follow these steps:

### Step 1: Rewrite the Integral
We can express \( \frac{1}{\cos^4 x} \) in terms of secant:
\[ I = \int \sec^4 x \, dx \]

### Step 2: Use the Identity for Secant
Recall the identity:
\[ \sec^2 x = 1 + \tan^2 x \]
Thus, we can express \( \sec^4 x \) as:
\[ \sec^4 x = (\sec^2 x)^2 = (1 + \tan^2 x)^2 \]
Expanding this gives:
\[ \sec^4 x = 1 + 2\tan^2 x + \tan^4 x \]

### Step 3: Split the Integral
Now we can split the integral:
\[ I = \int (1 + 2\tan^2 x + \tan^4 x) \, dx \]
This can be separated into three integrals:
\[ I = \int 1 \, dx + 2 \int \tan^2 x \, dx + \int \tan^4 x \, dx \]

### Step 4: Solve Each Integral
1. The first integral:
\[ \int 1 \, dx = x \]

2. The second integral:
To integrate \( \tan^2 x \), we use the identity \( \tan^2 x = \sec^2 x - 1 \):
\[ \int \tan^2 x \, dx = \int (\sec^2 x - 1) \, dx = \int \sec^2 x \, dx - \int 1 \, dx = \tan x - x \]
Therefore,
\[ 2 \int \tan^2 x \, dx = 2(\tan x - x) = 2\tan x - 2x \]

3. The third integral:
To integrate \( \tan^4 x \), we can use the identity \( \tan^4 x = \tan^2 x \cdot \tan^2 x \):
\[ \int \tan^4 x \, dx = \int (\sec^2 x - 1)^2 \, dx = \int (\sec^4 x - 2\sec^2 x + 1) \, dx \]
This requires integrating \( \sec^4 x \), which we already have as part of our original integral \( I \). We can express it as:
\[ \int \tan^4 x \, dx = \frac{1}{3} \tan^3 x - \tan x + x \]

### Step 5: Combine the Results
Now, we can combine all the results:
\[ I = x + (2\tan x - 2x) + \left(\frac{1}{3} \tan^3 x - \tan x + x\right) \]
Simplifying this gives:
\[ I = \frac{1}{3} \tan^3 x + \tan x - x + C \]
where \( C \) is the constant of integration.

### Final Result
Thus, the integral \( I \) is:
\[ I = \frac{1}{3} \tan^3 x + \tan x - x + C \]