Too innocent of an integral

Calculus Level 2

Let f(x)=x+ex1f(x) = x + e^x - 1, and let f1(x)f^{-1} (x) denote the inverse function of f(x)f(x) .

Then the integral e1+e2f1(x)dx \displaystyle \int_{e}^{1 + e^2} f^{-1} (x) \, dx evaluates to k0+k1e1+k2e2++knen, k_0 + k_1 e^1 + k_2 e^2 + \cdots + k_n e^n , where k0,k1,,knk_0, k_1, \ldots, k_n are rational numbers and e2.718e\approx 2.718 is Euler's number.

If the sum k0+k1+k2++knk_0 + k_1 + k_2 + \cdots + k_n can be expressed as AB,\frac{A}{B}, where AA and BB are coprime positive integers, what is A+B+n?A+B+n?

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