How do you integrate $t\coth(x t)/\sqrt{1-t^2}$ from $t = 0$ to $1$? How do you do this integral:
$$A=\int\limits_{0}^{1}\mathrm{d}t\frac{t}{\sqrt{1-t^2}}\coth(xt)$$
for a positive parameter x.
 A: For the further treatment of the integral a little trick is needed:  
$$\mathbf{A}\left( x\right) =\int_{0}^{1}\left( \int \frac{~t}{\sqrt{1-t^{2}}}%
\coth \left( x~t\right) dx\right) \;dt=\int_{0}^{1}\frac{\log \left( \sinh
\left( t~x\right) \right) }{\sqrt{1-t^{2}}}\;dt$$
where $\mathbf{A}\left( x\right) $ is the antiderivative of $A$.  We write $\mathbf{A}%
\left( x\right) $ in a series representation: 
$$\mathbf{A}\left( x\right) =\int_{0}^{1}\frac{1}{\sqrt{1-t^{2}}}\left(
t~x-\sum_{n\geq 1}\frac{\exp \left( -2n~t~x\right) }{n}\right) \;dt$$
Integration over $t$ leads to the generalized Schlömilch series
$$\mathbf{A}\left( x\right) =x-\frac{\pi }{2}T_{\nu ,\mu }^{I,L}\left(
x\right) $$
with
$$T_{\nu ,\mu }^{I,L}\left( y\right) =\sum_{n\geq 1}\frac{I_{\nu }\left(
n~y\right) -L_{\nu }\left( n~y\right) }{n^{\mu }}$$
for $y=2x$,$\;\nu =0$ and $\mu =1$. $I_{\nu }$ and  $L_{\nu }$ are the modified Bessel function of the first kind and modified Struve function. In the end we have: 
$$A\left( x\right) =1-\frac{\pi }{2}\left( T_{\nu ,\mu }^{I,L}\left( x\right)
\right) ^{\prime }$$
The Integral can further be simplified using the Sonin-Gubler formula as
discussed in detail in  POG´ANY
