\int \frac{\ln x+1}{(\ln x)^2(1+x\ln x)}\mathrm{d}x

To evaluate the integral \(\int \frac{\ln x + 1}{(\ln x)^2 (1 + x \ln x)} \, \mathrm{d}x\), we can use a substitution method. Let's proceed step by step.

First, let \(u = \ln x\). Then, \(du = \frac{1}{x} \, dx\), and therefore \(dx = x \, du = e^u \, du\). 

Rewrite the integral in terms of \(u\):
\[ x = e^u \]
\[ \ln x = u \]
\[ dx = e^u \, du \]

Now substitute these into the integral:
\[ \int \frac{u + 1}{u^2 (1 + e^u u)} e^u \, du \]

Simplify the integrand:
\[ \int \frac{u + 1}{u^2 (1 + u e^u)} e^u \, du = \int \frac{u + 1}{u^2 (u e^u + 1)} e^u \, du \]

\[ = \int \frac{u + 1}{u^3 e^u + u^2} e^u \, du = \int \frac{(u + 1) e^u}{u^3 e^u + u^2} \, du \]

\[ = \int \frac{u + 1}{u^3 + u^2 e^{-u}} \, du \]

Notice that this integrand can be split into two separate integrals:
\[ = \int \frac{u}{u^3 + u^2 e^{-u}} \, du + \int \frac{1}{u^3 + u^2 e^{-u}} \, du \]

This still looks quite complicated. Instead of splitting, let's simplify differently by breaking the fraction:
\[ = \int \left( \frac{1}{u^2} \cdot \frac{u}{u(1 + e^{-u})} \right) \, du + \int \left( \frac{1}{u^2} \cdot \frac{1}{1 + e^{-u}} \right) \, du \]

\[ = \int \frac{1}{u(1 + e^{-u})} \, du + \int \frac{1}{u^2 (1 + e^{-u})} \, du \]

Both integrals are still complex. To proceed further analytically would require advanced techniques or recognizing a special function. 

Another way to address it is by using numerical methods or approximation methods for more practical values.

However, we can use another substitution to simplify it more:
\[ t = \ln x \implies dt = \frac{1}{x} dx \]

\[ I = \int \frac{t + 1}{t^2 (1 + x t)} x \, dt = \int \frac{t + 1}{t^2 (1 + e^t t)} e^t \, dt \]

In some cases, integral tables or computer algebra systems (CAS) might be required to get the exact answer.

Would you like more details or have any questions? Here are some questions for you to consider:

1. Can you provide steps for another integral you are working on?
2. Do you need help with a specific technique in calculus?
3. Are there any topics in algebra or trigonometry you need clarification on?
4. Do you want to learn more about substitution methods in integration?
5. Is there a particular area of mathematics you're focusing on right now?

**Tip:** When integrating complex rational functions, always consider substitution and breaking down the integrand into simpler parts to make the integration process more manageable.

Learn how to evaluate the integral \(\int \frac{\ln x + 1}{(\ln x)^2 (1 + x \ln x)} \, \mathrm{d}x\) using substitution and simplification techniques. This problem involves complex rational functions and requires advanced understanding of integration methods.

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