Evaluate `int(dx)/(sqrt(1+e^(x)+e^(2x)))`

Explanation

To evaluate the integral

\[ I = \int \frac{dx}{\sqrt{1 + e^x + e^{2x}}} \]

we can follow these steps:

### Step 1: Substitution
Let \( e^x = \frac{1}{y} \). Then, differentiating both sides gives us:

\[ dx = -\frac{1}{y^2} dy \]

### Step 2: Rewrite the Integral
Substituting \( e^x \) and \( dx \) into the integral, we have:

\[ I = \int \frac{-\frac{1}{y^2} dy}{\sqrt{1 + \frac{1}{y} + \frac{1}{y^2}}} \]

### Step 3: Simplify the Denominator
The expression inside the square root simplifies as follows:

\[ 1 + \frac{1}{y} + \frac{1}{y^2} = \frac{y^2 + y + 1}{y^2} \]

Thus, we can rewrite the integral as:

\[ I = -\int \frac{y^2}{\sqrt{y^2 + y + 1}} \cdot \frac{1}{y^2} dy = -\int \frac{1}{\sqrt{y^2 + y + 1}} dy \]

### Step 4: Completing the Square
To simplify \( y^2 + y + 1 \), we complete the square:

\[ y^2 + y + 1 = \left(y + \frac{1}{2}\right)^2 + \frac{3}{4} \]

### Step 5: Rewrite the Integral
Now, substituting back into the integral gives:

\[ I = -\int \frac{1}{\sqrt{\left(y + \frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2}} dy \]

### Step 6: Use the Integral Formula
Using the formula for the integral

\[ \int \frac{1}{\sqrt{x^2 + a^2}} dx = \ln |x + \sqrt{x^2 + a^2}| + C \]

we can apply it here. Let \( x = y + \frac{1}{2} \) and \( a = \frac{\sqrt{3}}{2} \):

\[ I = -\ln \left| y + \frac{1}{2} + \sqrt{\left(y + \frac{1}{2}\right)^2 + \frac{3}{4}} \right| + C \]

### Step 7: Substitute Back for \( y \)
Recalling that \( y = \frac{1}{e^x} \), we substitute back:

\[ I = -\ln \left| \frac{1}{e^x} + \frac{1}{2} + \sqrt{\left(\frac{1}{e^x} + \frac{1}{2}\right)^2 + \frac{3}{4}} \right| + C \]

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

\[ I = -\ln \left( \frac{1 + \frac{1}{2} e^x + \sqrt{(1 + \frac{1}{2} e^x)^2 + \frac{3}{4} e^{2x}}}{e^x} \right) + C \]