integrate the following integral Can anyone help with integrating the following integral:
$$I=\int{\frac{\ln(\tan(x))}{\sin(x)+\cos(x)}}dx$$
I tried with mathematica,matlab, all wouldn't come out as a simple answer.
Just wonder if anyone can work out here.
Many thanks!
 A: You won't obtain a general answer since the integral is not defined. The tangens is negative on $(\frac{1}{2}\pi,\pi)$, $(\frac{3}{2}\pi,2\pi)$ and so on. More precisely it is negative in $(\frac{2n+1}{2}\pi,n\pi), n\in \mathbb{Z}$. But the logarithm is not defined for negative numbers! So the function over which you integrate is not defined everywhere.
So let us have a look where it is defined. That is $(n\pi,\frac{2n+1}{2}\pi), n\in \mathbb{Z}$. Let's have a special look at $(0,\frac{\pi}{2})$. The function is antisemetric to the point $\pi/4$ thus:
$$\int_0^{\frac{\pi}{2}}{\frac{\ln(\tan(x))}{\sin(x)+\cos(x)}}dx = 0$$
On the other side 
$$\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}{\frac{\ln(\tan(x))}{\sin(x)+\cos(x)}}dx \approx 0.792737,$$
which I obtained with Wolfram Alpha (It was probably numerically evaluated).
It might be possible to obtain an expression for the antiderivative
$$\int_{\frac{\pi}{4}}^{y}{\frac{\ln(\tan(x))}{\sin(x)+\cos(x)}}dx \text{, where } y\in (0,\tfrac{\pi}{2}) .$$
But I don't directly see how. Why are you interested in this integral? 
