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The expression you’ve provided is:

\[
\int \left( \frac{\arctan \frac{3}{x}}{1 + x^2} + 4^{\sin^x} \cdot \cos x \right) dx
\]

This involves two parts under the integral: 
1. \(\frac{\arctan \frac{3}{x}}{1 + x^2}\)
2. \(4^{\sin^x} \cdot \cos x\)

### Step 1: Breakdown of the first term
We have \(\frac{\arctan \frac{3}{x}}{1 + x^2}\). To integrate this, we can apply standard rules for integrating expressions involving arctangent functions and derivatives of trigonometric functions.

### Step 2: Handling the second term
The second term is \(4^{\sin^x} \cdot \cos x\), which is more complex due to the exponential and trigonometric interaction. This might require substitution or series expansion methods to find an exact solution.

### Would you like a detailed step-by-step solution for these terms?

∫ (arctan(3/x) / (1 + x^2) + 4^(sin^x) * cos(x)) dx

Learn how to solve a complex integral involving arctan(3/x) over (1 + x^2) and 4^(sin^x) multiplied by cos(x). This problem covers key calculus concepts including the integration of inverse trigonometric and exponential functions. Suitable for university-level calculus students.

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