The Laplace transform of the digamma function: An integral due to Glasser, Manna and Oloa

The definite integral M ( a ) := 4 π ∫ 0 π / 2 x 2 d x x 2 + ln 2 ⁡ ( 2 e − a cos ⁡ x ) \begin{equation*} M(a):= \frac {4}{\pi } \int _{0}^{\pi /2} \frac {x^{2} \, dx } {x^{2} + \ln ^{2}( 2 e^{-a} \cos x ) }\end{equation*} is related to the Laplace transform of the digamma function L ( a ) := ∫ 0 ∞ e − a s ψ ( s + 1 ) d s , \begin{equation*} L(a) := \int _{0}^{\infty } e^{-a s} \psi (s+1) \, ds, \end{equation*} by M ( a ) = L ( a ) + γ / a M(a) = L(a) + \gamma /a when a > ln ⁡ 2 a > \ln 2 . Certain analytic expressions for M ( a ) M(a) in the complementary range, 0 > a ≤ ln ⁡ 2 0 > a \leq \ln 2 , are also provided.