Evaluate `inte^(3logx)(x^(4)+1)^(-1)dx`

Explanation

To evaluate the integral \( \int e^{3 \log x} \cdot (x^4 + 1)^{-1} \, dx \), we can follow these steps:

### Step 1: Simplify the integrand
We start by simplifying \( e^{3 \log x} \). Using the property \( e^{\log a} = a \), we have:

\[ e^{3 \log x} = e^{\log(x^3)} = x^3 \]

Thus, the integral becomes:

\[ \int \frac{x^3}{x^4 + 1} \, dx \]

### Step 2: Rewrite the integral
We can rewrite the integral as follows:

\[ \int \frac{x^3}{x^4 + 1} \, dx \]

### Step 3: Use substitution
Next, we can use the substitution \( t = x^4 + 1 \). Then, we differentiate \( t \):

\[ dt = 4x^3 \, dx \quad \Rightarrow \quad dx = \frac{dt}{4x^3} \]

### Step 4: Substitute in the integral
Now, substituting \( t \) and \( dx \) into the integral gives:

\[ \int \frac{x^3}{t} \cdot \frac{dt}{4x^3} = \int \frac{1}{4t} \, dt \]

### Step 5: Integrate
Now we can integrate:

\[ \int \frac{1}{4t} \, dt = \frac{1}{4} \ln |t| + C \]

### Step 6: Substitute back
Now we substitute back \( t = x^4 + 1 \):

\[ \frac{1}{4} \ln |x^4 + 1| + C \]

### Final Answer
Thus, the final answer is:

\[ \int e^{3 \log x} \cdot (x^4 + 1)^{-1} \, dx = \frac{1}{4} \ln(x^4 + 1) + C \]