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Why this function is elementary and its pair is not?

Why $$f(x)=\frac{2 \zeta (2)}{\pi ^2}+\frac{6 \zeta(4) x^2}{\pi^4}+\frac{10 \zeta(6) x^4}{\pi^6}+\frac{14 \zeta(8) x^6}{\pi^8}+\cdots$$ is elementary while $$g(x)=\frac{4 \zeta (3)x}{\pi ...
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Question about elementary and nonelementary functions.

Let $E(z)$ be an entire elementary function of (complex) $z$ and $N(z)$ be an entire nonelementary function of (complex) $z$. $e^{N(z)}$$N'(z) = E(z)$ The ' means derivative with respect to $z$. ...