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Is there a closed form for $\int_0^\infty\frac{\tanh^3(x)}{x^2}dx$?

Following the suggestion I made in a comment, the integral can be rewritten as the contour integral $$I_{3,2} = \frac{1}{2\pi i} \oint \frac{\operatorname{tanh}^3 z}{z^2} \log(-z) \, dz ,$$ where ...
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Is there a closed form for $\int_0^\infty\frac{\tanh^3(x)}{x^2}dx$?

Rewrite the integrand and apply Taylor expansion to $\frac1{(1+e^{-2x})^3}$ so that $$\frac{\tanh^3x}{x^2}=\sum_{j\geq0}(-1)^j\binom{j+2}2 \frac{(1-e^{-2x})^3}{x^2}\binom{j+2}2e^{-2jx}.$$ Integrate ...
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On 12 cfracs: for Catalan's $K$, Gieseking's $\kappa$, and $\pi^2$, $\pi^3$, plus three for $\zeta(3)$ using Zagier's "six sporadic sequences"

We have $$C_2(-17,-6,-72)=-(5/8)L(\chi_{-3},2)$$ and $$C_2(10,3,-9)=(1/2)L(\chi_{-3},2)$$ so both are proportional to what you call Gieseking's constant but which is simply the value at 2 of the L ...
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Closed Form for $\sum\limits_{n=-2a}^\infty(n+a){2a\choose-n}^4,~a\not\in\mathbb Z$

Some experimentation suggests that the second series is given by $$S_4^-(a) = \sum_{n=-2a}^\infty (n+a) \binom{2a}{-n}^4 = \frac{1}{4\cos(2\pi a) \,\Gamma(2a+1)^2\,\Gamma(-4a)},$$ which agrees with ...
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When can an invertible function be inverted in closed form?

Closed-form functions need the definition which set of functions is allowed to represent the function. Take e.g. the algebraic definition of the Elementary functions by Liouville and Ritt (Wikipedia: ...
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On Zagier's missing continued fraction with multiple limits?

Set $Q=(1/2)L(\chi_{-3},2)$ (related to your Gieseking constant) and $P=2\pi^2/81$. The limits are almost certainly (not proved), \begin{align} \lim_{m\to\infty}C_2(6m+0) &= -Q\\ \lim_{m\to\infty}...
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Partition numbers as the specific sums of the A161511

If $2m=2^{j_1}+2^{j_2}+\ldots+2^{j_k}$ for $0<j_1<j_2<\ldots<j_k$, then $\ell(m)=j_k-1$, $a(m)=j_1+(j_2-1)+(j_3-2)+\ldots+(j_k-(k-1))$, as the $i$-th (from the left) 1 in the binary ...
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how to solve $\sum_{i=0}^n (x-\mu_i)e^{-(x-\mu_i)^2} = 0$
There always exists at least one solution: If $x >\mu_i$ for all $i$, then each of the terms in the sum is positive and if $x < \mu_i$ for all $i$, then each of the terms in the sum is negative. ...