We are tasked with solving the integral:

$I = \int \frac{1}{x^2(1+x^2)} \, dx$

Step 1: Decompose the integrand

We can attempt to simplify the integrand by using partial fraction decomposition. We want to express:

$\frac{1}{x^2(1+x^2)} = \frac{A}{x} + \frac{B}{x^2} + \frac{Cx + D}{1 + x^2}$

Step 2: Multiply both sides by the common denominator $x^2(1+x^2)$

$1 = A x(1 + x^2) + B(1 + x^2) + (Cx + D) x^2$

Expanding both sides:

$1 = A x + A x^3 + B + B x^2 + C x^3 + D x^2$

Now, collect like terms:

$1 = (A + C) x^3 + (B + D) x^2 + A x + B$

Step 3: Set up a system of equations

For the equation to hold for all $x$, the coefficients of the powers of $x$ on both sides must match. The left-hand side has no $x^3$, $x^2$, or $x$ terms, and the constant is 1. This gives us the system of equations:

1. $A + C = 0$
2. $B + D = 0$
3. $A = 0$
4. $B = 1$

Step 4: Solve the system

From equation (3), we have $A = 0$. Substituting into equation (1), we get $C = 0$. From equation (4), we have $B = 1$, and from equation (2), we find $D = -1$.

Thus, the partial fraction decomposition is:

$\frac{1}{x^2(1+x^2)} = \frac{1}{x^2} - \frac{1}{1+x^2}$

Step 5: Integrate each term

Now we can integrate term by term:

$I = \int \left( \frac{1}{x^2} - \frac{1}{1+x^2} \right) dx$

1. $\int \frac{1}{x^2} \, dx = -\frac{1}{x}$
2. $\int \frac{1}{1+x^2} \, dx = \tan^{-1}(x)$

Thus, the integral is:

$I = -\frac{1}{x} - \tan^{-1}(x) + C$

where $C$ is the constant of integration.

Final Answer:

$\int \frac{1}{x^2(1+x^2)} \, dx = -\frac{1}{x} - \tan^{-1}(x) + C$

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Would you like to see more detailed steps for any part of this solution, or do you have further questions? Here's a tip:

Tip: When dealing with complex rational functions, always consider partial fraction decomposition, as it can simplify the process of integration into familiar forms.

Also, here are some related questions you might find useful:

1. How do you integrate functions of the form $\int \frac{1}{x^2 + a^2} \, dx$?
2. What happens if you try a trigonometric substitution for integrals involving $1 + x^2$?
3. How can you apply the method of substitution for integrals with $x^2$ in the denominator?
4. What’s the difference between using partial fraction decomposition and completing the square in integration?
5. How would you solve a similar integral where the denominator is a cubic term?