The given image contains a mathematical integral and its evaluation. Let me analyze and rewrite it in LaTeX for clarity:

The Problem

$\int_{0}^{\infty} \frac{\pi x \Big( \pi x \csch(\pi x) \Big( \pi^2 x^2 + 6\pi x \Big(\coth(\pi x) + \csch^2(\pi x) + 6\Big) - 3\Big(\pi^2 x^2 - 2\Big)\sech(\pi x) \Big) - 24 \tanh(\pi x) \Big)}{x^5} \, dx$

The Solution (as given)

$= 12 \pi^3 G + 2 \log(2) \pi^4 - \frac{\pi^5}{3} - 744 \zeta(5) + \frac{\pi}{8} \psi^{(3)} \left( \frac{1}{4} \right)$

Where:

• $G$ is the Catalan's constant,
• $\zeta(5)$ is the Riemann zeta function at $5$,
• $\psi^{(3)}$ is the polygamma function of order $3$.

Would you like me to:

1. Provide a detailed step-by-step solution for evaluating this integral?
2. Explain the components of the solution (e.g., Catalan's constant, Riemann zeta, polygamma)?
3. Verify the solution or simplify specific steps for clarity?
4. Analyze or explain the functions involved in the integrand?
5. Explore related integrals or generalizations?

Tip:

Breaking the integral into simpler parts and addressing them step by step often makes such complex problems more manageable.