We state two lemmata first.
Lemma 1: Let . It holds that
Proof: A standard proof can be found through Fourier Series. One can expand in Fourier series the function . Another way to prove the identity is to begin from the Weierstrass product , that is . Taking log on both sides we have that
Differentiating we have
Lemma 2: It holds that
Proof: Just differentiate the above identity.
Apply the above lemmata , we have for the initial integral that