To solve the integral $\int \tanh\left(1 + t^2\right) \, dt$, we'll proceed with a substitution method to simplify the problem. Let's go through the steps:

Given: $\int \tanh(1 + t^2) \, dt$

Step 1: Let $u = 1 + t^2$

Then, differentiate $u$ with respect to $t$: $du = 2t \, dt \implies dt = \frac{du}{2t}$

Step 2: Substitute $u$ and $dt$ into the integral

$\int \tanh(u) \, \frac{du}{2t}$

However, notice that we still have $t$ in the denominator, which is expressed in terms of $u$. Since $u = 1 + t^2$, we can solve for $t$:

$t^2 = u - 1 \implies t = \sqrt{u - 1}$

Thus, the integral becomes: $\int \tanh(u) \, \frac{du}{2\sqrt{u - 1}}$

Step 3: Simplify the integral

The integral: $\int \frac{\tanh(u)}{2\sqrt{u - 1}} \, du$

This integral is non-trivial and may not have a simple closed-form solution using elementary functions. In many cases, such integrals can be evaluated using numerical methods or expressed in terms of special functions.

Conclusion

The integral $\int \tanh(1 + t^2) \, dt$ is complex and does not simplify easily using elementary methods. For a more detailed analysis, we might need to resort to numerical integration or special functions.

Would you like to see a numerical approximation for specific bounds, or do you need help with a different integral or mathematical concept?

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Here are some related questions you might be interested in:

1. How to solve integrals involving hyperbolic functions?
2. What are some common techniques for integrating complex functions?
3. Can you provide examples of using special functions in integrals?
4. How to approach integrals without closed-form solutions?
5. What are some methods for numerical integration?

Tip: When dealing with integrals of complex functions, substitutions and transformations can often simplify the problem, even if they don't always lead to a closed-form solution.