Simple Yet Challenging Integral

Calculus Level 3

Evaluate the definite integral \(\displaystyle{\int _{ 0 }^{ \frac{\pi}{4}}{ \ln { (\tan { x+1)\, dx } } } .}\) The answer must be a decimal in terms of \(\pi\). For example, if the calculation result is \(2.654\pi,\) then the answer should be \(2.654\).