Calculus Level 5

Define

$f(x) = \dfrac{1}{2} \tan\left(\dfrac{x}{2}\right) + \dfrac{1}{4} \tan\left(\dfrac{x}{4}\right) + \dfrac{1}{8} \tan\left(\frac{x}{8}\right) +\cdots.$

Then $$f\left(\dfrac{3\pi}{8}\right)$$ can be written as $\dfrac{a}{b\pi^{c}} + d - \sqrt{e}$

If $$\dfrac{a}{b}$$ is in simplest form i.e. $$\gcd(a,b)=1$$, and $$a,b,c,d,e\in\mathbb N$$, find the value of $$a + b + c + d + e$$.

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