`int(secx" cosec "x)/(log(tanx))dx`

Answer

To solve the integral \( \int \frac{\sec x \csc x}{\log(\tan x)} \, dx \), we will follow a systematic approach using substitution.

### Step-by-Step Solution:

1. **Substitution**: Let \( t = \log(\tan x) \).
- Then, we need to find \( dt \).
- The derivative of \( t \) with respect to \( x \) is:
\[ dt = \frac{1}{\tan x} \cdot \sec^2 x \, dx \]
- Rearranging gives:
\[ dx = \frac{\tan x}{\sec^2 x} dt \]

2. **Express \( \tan x \)**: We know that:
\[ \tan x = \frac{\sin x}{\cos x} \]
- Thus, we can express \( \frac{1}{\tan x} \) as:
\[ \frac{1}{\tan x} = \frac{\cos x}{\sin x} = \csc x \cos x \]

3. **Substituting back**: Now, substituting \( dx \) in the integral:
\[ \int \frac{\sec x \csc x}{\log(\tan x)} \, dx = \int \frac{\sec x \cdot \csc x \cdot \frac{\tan x}{\sec^2 x}}{\log(\tan x)} \, dt \]
- Simplifying this gives:
\[ = \int \frac{\sec x \cdot \csc x \cdot \frac{\sin x}{\cos x}}{\log(\tan x)} dt \]
- Since \( \sec x = \frac{1}{\cos x} \) and \( \csc x = \frac{1}{\sin x} \), we can simplify further:
\[ = \int \frac{1}{\log(\tan x)} dt \]

4. **Integrating**: The integral of \( \frac{1}{t} \) is:
\[ \int \frac{1}{t} dt = \log |t| + C \]
- Substituting back \( t = \log(\tan x) \):
\[ = \log |\log(\tan x)| + C \]

### Final Answer:
Thus, the final result of the integral is:
\[ \int \frac{\sec x \csc x}{\log(\tan x)} \, dx = \log |\log(\tan x)| + C \]