In a recent lecture a professor told a story about the integral below. Lobachevsky calculated this integral at first time incorrectly. Following the publication of the integral, Ostrogradsky sent a letter with correct answer to Lobachevsky.
What is the right answer?
$I(a)=\int_{0}^{\infty}\frac{e^x-e^{-x}}{e^{2x}+e^{-2x}+2cos(2a)}xdx;0\leq a \leq \pi$