Verified answer

Is that (e^(x) + 1) / (e^(x) - 1) or e^(x) + (e^(x) - 1)^(-1)?

I'll presume the latter

y = ln(e^(x) + (e^(x) - 1)^(-1))

e^(y) = e^(x) + (e^(x) - 1)^(-1)

e^(y) * dy = e^(x) * dx - (e^(x) * dx) / (e^(x) - 1)^2

e^(y) * dy = e^(x) * dx * (1 - 1/(e^(x) - 1)^2)

dy/dx = e^(x) * (((e^(x) - 1)^2 - 1) / (e^(x) - 1)^2) / e^(y)

dy/dx = e^(x) * (e^(2x) - 2 * e^(x) + 1 - 1) / (e^(y) * (e^(x) - 1)^2)

dy/dx = e^(x) * (e^(2x) - 2 * e^(x)) / (e^(y) * (e^(x) - 1)^2)

dy/dx = e^(2x) * (e^(x) - 2) / ((e^(x) + (e^(x) - 1)^(-1)) * (e^(x) - 1)^2)

dy/dx = e^(2x) * (e^(x) - 2) / ((e^(x) * (e^(x) - 1) + 1) * (e^(x) - 1)^2 / (e^(x) - 1))

dy/dx = e^(2x) * (e^(x) - 2) / ((e^(2x) - e^(x) + 1) * (e^(x) - 1))

dy/dx = e^(2x) * (e^(x) - 2) / (e^(3x) - e^(2x) - e^(2x) + e^(x) + e^(x) - 1)

dy/dx = e^(2x) * (e^(x) - 2) / (e^(3x) - 2 * e^(2x) + 2 * e^(x) - 1)

sqrt(1 + (dy/dx)^2) * dx =>

sqrt(1 + e^(4x) * (e^(2x) - 4 * e^(x) + 4) / (e^(3x) - 2 * e^(2x) + 2 * e^(x) - 1)^2) * dx =>

sqrt((e^(3x) - 2 * e^(2x) + 2 * e^(x) - 1)^2 + e^(6x) - 4 * e^(5x) + 4 * e^(4x)) * dx / (e^(3x) - 2 * e^(2x) + 2 * e^(x) - 1)

(e^(3x) - 2 * e^(2x) + 2 * e^(x) - 1)^2 =>

e^(6x) - 4 * e^(5x) + 4 * e^(4x) - 2 * e^(3x) + 4 * e^(4x) - 8 * e^(3x) + 4 * e^(2x) + 4 * e^(2x) - 4 * e^(x) + 1 =>

e^(6x) - 4 * e^(5x) + 8 * e^(4x) - 10 * e^(3x) + 8 * e^(2x) - 4 * e^(x) + 1

sqrt((e^(3x) - 2 * e^(2x) + 2 * e^(x) - 1)^2 + e^(6x) - 4 * e^(5x) + 4 * e^(4x)) * dx / (e^(3x) - 2 * e^(2x) + 2 * e^(x) - 1)

sqrt(e^(6x) + e^(6x) - 4 * e^(5x) - 4 * e^(5x) + 8 * e^(4x) + 4 * e^(4x) - 10 * e^(3x) + 8 * e^(2x) - 4 * e^(x) + 1) * dx / (e^(3x) - 2 * e^(2x) + 2 * e^(x) - 1)

sqrt(2 * e^(6x) - 8 * e^(5x) + 12 * e^(4x) - 10 * e^(3x) + 8 * e^(2x) - 4 * e^(x) + 1) * dx

If there's a way to do this, then it's beyond me.

Now, let's suppose you meant (e^(x) + 1) / (e^(x) - 1) instead

y = ln((e^(x) + 1) / (e^(x) - 1))

e^(y) = (e^(x) + 1) / (e^(x) - 1)

e^(y) = (e^(x) - 1 + 2) / (e^(x) - 1)

e^(y) = (e^(x) - 1) / (e^(x) - 1) + 2 / (e^(x) - 1)

e^(y) = 1 + 2 / (e^(x) - 1)

e^(y) * dy = 0 + ((e^(x) - 1) * 0 - 2 * (e^(x) * dx)) / (e^(x) - 1)^2

e^(y) * dy = -2 * e^(x) * dx / (e^(x) - 1)^2

dy/dx = -2 * e^(x) / (e^(y) * (e^(x) - 1)^2)

dy/dx = -2 * e^(x) / (((e^(x) + 1) / (e^(x) - 1)) * (e^(x) - 1)^2)

dy/dx = -2 * e^(x) / ((e^(x) + 1) * (e^(x) - 1))

dy/dx = -2 * e^(x) / (e^(2x) - 1)

sqrt(1 + (dy/dx)^2) * dx

sqrt(1 + 4 * e^(2x) / (e^(2x) - 1)^2) * dx

sqrt((1 * (e^(2x) - 1)^2 + 4 * e^(2x)) / (e^(2x) - 1)^2) * dx

sqrt(e^(4x) - 2 * e^(2x) + 4 * e^(2x) + 1) * dx / (e^(2x) - 1)

sqrt(e^(4x) + 2 * e^(2x) + 1) * dx / (e^(2x) - 1)

sqrt((e^(2x) + 1)^2) * dx / (e^(2x) - 1)

(e^(2x) + 1) * dx / (e^(2x) - 1)

(e^(2x) - 1 + 2) * dx / (e^(2x) - 1)

(e^(2x) - 1) * dx / (e^(2x) - 1) + 2 * dx / (e^(2x) - 1)

dx + 2 * dx / (e^(2x) - 1)

e^(2x) - 1 = sec(t)^2 - 1

e^(x) = sec(t)

e^(x) * dx = sec(t) * tan(t) * dt

sec(t) * dx = sec(t) * tan(t) * dt

dx = tan(t) * dt

dx + 2 * dx / (e^(2x) - 1)

dx + 2 * tan(t) * dt / (sec(t)^2 - 1)

dx + 2 * tan(t) * dt / tan(t)^2

dx + 2 * dt / tan(t)

dx + 2 * dt / (sin(t) / cos(t))

dx + 2 * cos(t) * dt / sin(t)

u = sin(t)

du = cos(t) * dt

dx + 2 * du / u

Integrate

x + 2 * ln|u| + C

x + 2 * ln|sin(t)| + C

e^(x) = sec(t)

e^(-x) = cos(t)

e^(-2x) = cos(t)^2

e^(-2x) = 1 - sin(t)^2

sin(t)^2 = 1 - e^(-2x)

sin(t) = (1 - e^(-2x))^(1/2)

x + 2 * ln|sin(t)| + C

x + 2 * ln|(1 - e^(-2x))^(1/2)| + C

x + 2 * (1/2) * ln|1 - e^(-2x)| + C

x + ln|1 - e^(-2x)| + C

I'm thinking that this is what you were going for

From x = a to x = b

(b - a) + ln|1 - e^(-2b)| - ln|1 - e^(-2a)|

(b - a) + ln|(e^(2b) - 1) / e^(2b)| - ln|(e^(2a) - 1) / e^(2a)|

(b - a) + ln|e^(2a) * (e^(2b) - 1) / (e^(2b) * (e^(2a) - 1)|

(b - a) + ln(e^(2a)) + ln(e^(b) - 1) + ln(e^(b) + 1) - ln(e^(2b)) - ln(e^(a) - 1) - ln(e^(a) + 1)

b - a + 2a + ln(e^(2b) - 1) - 2b - ln(e^(2a) - 1)

a - b + ln((e^(2b) - 1) / (e^(2a) - 1))