Ensure the innermost function is greater than zero:

Simplify the inequality

Solve the quadratic inequality:

This inequality indicates that must lie between the roots, giving the interval

With the domain of identified as , we calculate the definite integral over :

Calculate the Integral:
Given:

For , we compute:

Computation of Each Integral Segment:

Summing these, we have:

Conclusion:

The values for , and are , and , with the greatest common divisor of these numbers being 1. Therefore, adding them together gives:

──────────────────────────────────────────────────────────────────────────────────────────