Algebraic manipulation of integral

[dirk_mec1]

Homework Statement

$$\int_0^{\frac{\pi}{2}} \frac{\cos(x)}{1+\sin^2(x)} \ln(1+\cos(x))\ dx$$

The Attempt at a Solution

Can anyone give me a hint as to what to do?

 

[Cyosis]

Using y=sin(x) followed by partial integration and some algebraic manipulation you can get to the following integral:

$$\int_0^1 \frac{\arctan(y)}{y\sqrt{1-y^2}}\ dy - \int_0^1 \frac{\arctan{y}}{y}\ dy$$

You know the first integral from a previous question you asked and the second integral is equal to Catalan's constant.