By methods of differentiation under the integral sign Could this result be proven by methods of differentiation under the integral sign? I only take interest in differentiation way.
$$\int_0^\infty \frac{\ln(\tan^2 (ax))}{1+x^2}dx = \pi\ln(\tanh(a))$$

Actually using the derivative really simplifies the problem. After that and a simple substitution one obtain:
$$
J'(a) = 4 \int_0^\infty \frac{t}{\sin t} \frac{dt}{t^2+4a^2}
$$
The last integral can be evaluated by means of complex analysis:
$$
J'(a) = \frac{2\pi}{\sinh 2a}
$$
I guess this is what they really wanted in this problem.
