Too innocent of an integral

Calculus Level 3

Let \(f(x) = x + e^x - 1\), and let \(f^{-1} (x) \) denote the inverse function of \(f(x) \).

Then the integral \( \displaystyle \int_{e}^{1 + e^2} f^{-1} (x) \, dx \) evaluates to \[ k_0 + k_1 e^1 + k_2 e^2 + \cdots + k_n e^n ,\] where \(k_0, k_1, \ldots, k_n\) are rational numbers and \(e\approx 2.718\) is Euler's number.

If the sum \(k_0 + k_1 + k_2 + \cdots + k_n\) can be expressed as \(\frac{A}{B},\) where \(A\) and \(B\) are coprime positive integers, what is \(A+B+n?\)

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