As Little Work As Possible

Calculus Level 5

A particle can move on an infinite two-dimensional plane, which is given an $$Oxy$$-coordinate system. As the particle moves on the plane, when it is at the point with coordinates $$(x,y)$$, it experiences a resistance force equal to $$e^{-y}$$ Newtons in a direction opposite to that of motion of the particle. Thus the work done moving the particle along a particular path $$C$$ is $\int_C e^{-y}\,ds$ Joules, where $$s$$ represents the arc-length, measured in metres.

The particle has to be moved smoothly from the point $$A=(0,0)$$ to the point $$B=\big(\tfrac12\pi,1\big)$$, in such a way that the $$x$$-coordinate of its position never decreases. Thus a possible path from $$A$$ to $$B$$ is of the form $$(x,y(x))$$, where $$y\,:\, \left[0,\tfrac12\pi\right] \to \mathbb{R}$$ is a smooth function such that $$y(0) = 0$$ and $$y\big(\tfrac12\pi\big) = 1$$.

Of all the possible paths from $$A$$ to $$B$$ there is one for which the work done against the resistance force in moving the particle from $$A$$ to $$B$$ is as small as possible. The length of this path can be shown to be $P \tanh^{-1}\left( \frac{\sqrt{e^Q + R}}{Se + T}\right),$ where $$P,Q,R,S,T$$ are positive integers. What is the value of $$P+Q+R+S+T$$?

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