I hope there will be no convergence issues (eighteenth integral)

Calculus Level 5

\[ \displaystyle {\huge \int_{0}^{1} } \left( x + \dfrac{x}{x+ \dfrac{x}{x+\dfrac{x}{x+...}}} \right) \, dx \]

If the above integral can be represented in the form

\[ \dfrac{a}{b} + \dfrac{c \sqrt{d}}{f} - g \ln \left( \dfrac{h + \sqrt{j}}{k}\right) \]

where

  • \( \gcd(a, b) = \gcd(c, f) = \gcd(h, k) = 1 \)
  • \(a, b, c, d, f, g, h, j, k \) are all positive integers
  • \( d, j \) are square-free
  • \( g = 2\)

then find \( a+b+c+d+f+g+h+j+k \).

Bonus: What special number is contained somewhere in the answer to this problem?

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