I was wondering if there existed a closed form for $$\int \tan(x)\operatorname{tanh}(x), \operatorname{dx}$$ I don't think this integral has a closed form, but could it be evaluated over some points $a$ and $b$?
Note that solving the integral above is the same as solving $$\int\frac{i (e^{-i x}-e^{i x}) (e^x-e^{-x})}{(e^{-i x}+e^{i x}) (e^{-x}+e^x)}\operatorname{dx}$$ which may be an easier task.
By using the formula http://upload.wikimedia.org/wikipedia/en/math/7/2/a/72a1058ad2087aec467af24bddcf9479.png, we have $\int\tan x\tanh x~dx=x\sum\limits_{n=1}^\infty\sum\limits_{m=1}^{2^n-1}\dfrac{(-1)^{m+1}}{2^n}\tan\dfrac{mx}{2^n}\tanh\dfrac{mx}{2^n}+C$