I want to calculate the antiderivative of $x \mapsto e^x \coth(x)$ for $x > 0$. Wolfram-Alpha tells me that it is equal to $x \mapsto e^x - 2 \operatorname{artanh}(e^x)$. However, this is obviously only defined for $x < 0$.
2026-03-31 09:22:16.1774948936
Antiderivative of $e^x \coth(x)$?
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HINT: We can rewrite this function as $$x\mapsto e^x\coth(x) = \frac{e^x+\tfrac{1}{e^x}}{e^x-\tfrac{1}{e^x}}\cdot e^x$$ Can you recognize this function as taking the form $R(g(x))\cdot g'(x)$ for some function $g$ and rational function $R$, and then find the antiderivative of the rational function $R$ and use the chain rule?