This problem requires a careful application of algebraic factorization, trigonometric identities for inverse tangent, and standard integration techniques. The key is to strategically manipulate the integrand to simplify it into forms that are easily integrable.

1. Key Concepts and Formulas Used:

• Algebraic Factorization:
  • $a^2 + b^2 = (a+b)^2 - 2ab$
  • $a^6 + b^6 = (a^2)^3 + (b^2)^3 = (a^2+b^2)(a^4-a^2b^2+b^4)$
• Integral of $\frac{1}{x^2+a^2}$:
  • $\int \frac{1}{