$$I(a)= \int _0^ \infty tanh \pi x( \frac{1}{x} - \frac{x}{ x^{2} + a^{2} } )dx,*a \succeq 0:coshx= \prod _{k=1} ^ \infty (1+ \frac{4 x^{2} }{ (2k+1)^{2} \pi ^{2} } ),*tanhx=8x \sum _{k=1} ^ \infty \frac{1}{ \pi ^{2} (2k-1)^{2} +4x^{2} } \Longrightarrow I(a)= \frac{1}{ \pi a^{2} } \sum _ {k=1} ^ \infty \int _0^ \infty (\frac{8x}{4 x^{2} + (2k-1)^{2} } )( \frac{1}{x} - \frac{x}{ x^{2} + a^{2} } )dx,I_1= \int _0^ \infty \frac{8}{4 x^{2} + (2k-1)^{2} }dx= \frac{4}{2k-1} Arctan( \frac{2x}{2k-1} ) | _0^ \infty = \frac{2 \pi }{2k-1} ,I_2= \int _0^ \infty \frac{8 x^{2} }{(4 x^{2}+ (2k-1)^{2} )( x^{2} + a^{2} )}dx= \int _0^ \infty ( \frac{8 a^{2} }{(4 a^{2} - (2k-1)^{2} )( x^{2} + a^{2} )} - \frac{8 (2k-1)^{2} }{(4 a^{2} - (2k-1)^{2} )(4 x^{2}+ (2k-1)^{2} )} ) dx= \frac{2 \pi }{2k+2a-1} ,I(a)= \sum _{k=1} ^ \infty( \frac{2}{2k-1} - \frac{2}{2k+2a-1} ) :*z \geq 0 ,\psi (z+1)=- \gamma + \sum _{k=1} ^ \infty ( \frac{1}{n} - \frac{1}{n+z} ) \Longrightarrow ?= \psi (a+ \frac{1}{2} ) -\psi ( \frac{1}{2} )= \frac{1}{ a^{2} } ( \psi (a+ \frac{1}{2} )- \psi ( \frac{1}{2} )$$
