The given expression appears to be a combination of infinite series involving sine functions and some constants. Let’s analyze and simplify it step-by-step.

Step-by-Step Simplification:

1. Separate Terms Inside the Brackets: $3 \left[ \frac{L^2}{3} \left( \sum_{n=1}^{\infty} \frac{-2(-1)^n L}{\pi n} \sin\left(\frac{n \pi x}{L}\right) \right) + \frac{4L^3}{\pi^3} \sum_{n=1}^{\infty} \frac{(-1)^n}{n^3} \sin\left(\frac{n \pi x}{L}\right) \right]$

2. Distribute the Constant Outside: $= L^2 \sum_{n=1}^{\infty} \frac{-2(-1)^n L}{\pi n} \sin\left(\frac{n \pi x}{L}\right) + \frac{12 L^3}{\pi^3} \sum_{n=1}^{\infty} \frac{(-1)^n}{n^3} \sin\left(\frac{n \pi x}{L}\right)$

3. Combine Like Terms: Since both terms are sums over $\sin\left(\frac{n \pi x}{L}\right)$, we can combine them: $= \sum_{n=1}^{\infty} \left( \frac{-2(-1)^n L^3}{\pi n} + \frac{12 L^3 (-1)^n}{\pi^3 n^3} \right) \sin\left(\frac{n \pi x}{L}\right)$

4. Factor Out Common Terms: $= L^3 \sum_{n=1}^{\infty} \left( \frac{-2(-1)^n}{\pi n} + \frac{12 (-1)^n}{\pi^3 n^3} \right) \sin\left(\frac{n \pi x}{L}\right)$

5. Simplify the Coefficients: The series now simplifies to: $= L^3 \sum_{n=1}^{\infty} (-1)^n \left( \frac{-2}{\pi n} + \frac{12}{\pi^3 n^3} \right) \sin\left(\frac{n \pi x}{L}\right)$

   Breaking down each term, we obtain: $L^3 \sum_{n=1}^{\infty} (-1)^n \left( -\frac{2}{\pi n} + \frac{12}{\pi^3 n^3} \right) \sin\left(\frac{n \pi x}{L}\right)$

This is the simplified form of the expression in terms of an infinite series. To further simplify this into a closed form, we’d need to evaluate these series specifically, which could result in more simplified trigonometric expressions, depending on boundary conditions or specific functions involved.

Would you like to go further into evaluating these series or explore a different aspect of the expression?

Further Questions:

1. What specific values of $x$ and $L$ are you interested in evaluating for this function?
2. Are there specific boundary conditions or intervals for $x$ and $L$ you'd like to explore?
3. Would you like to see this simplified into a specific trigonometric or Fourier series form?
4. Do you need help with the convergence of the series?
5. Are there any applications or contexts where you will be using this expression?

Tip:

When working with infinite series, knowing common series expansions and their properties can greatly help in simplifying or evaluating terms faster.