<s>[INST] $\int_{0}^{\pi\over 2}\cos(x)\ln(\tan{x}){\mathrm dx\over 1+\tan(x)}=\int_{0}^{\pi\over 2}\cos(3x)\ln(\tan{x}){\mathrm dx\over 1+\tan(x)}=-\ln{2}?$ 
$$I=\int_{0}^{\pi\over 2}\cos(x)\ln(\tan{x}){\mathrm dx\over 1+\tan(x)}$$
$$J=\int_{0}^{\pi\over 2}\cos(3x)\ln(\tan{x}){\mathrm dx\over 1+\tan(x)}$$

How to show that $I=J=-\ln{2}?$

For Integral $I$
${\cos{x}\over 1+\tan{x}}={1-\sin^2{x}\over \sin{x}+\cos{x}}$
$$I=\int_{0}^{\pi\over 2}{\ln{\tan{x}}}{\mathrm dx\over \sin{x}+\cos{x}}-\int_{0}^{\pi\over 2}{\sin^2{x}}\ln(\tan{x}){\mathrm dx\over \sin{x}+\cos{x}}=I_1-I_2$$
$$$$
$u=\tan{x}$ then $\mathrm dx=\cos^2{x}\mathrm du$
$$I_1=\int_{0}^{\infty}{\ln{u}}{\mathrm du\over (1+u)\sqrt{1+u^2}}$$
$$I_2=\int_{0}^{\infty}{u^2\ln{u}}{\mathrm du\over (1+u)(1+u^2)\sqrt{1+u^2}}$$
 [/INST] As an alternative to the solution provided by Jack D'Aurizio, I will find a value to the integral $I$ by picking up where you left off.
For the first of your integrals $I_1$, if we write
$$I_1 = \int^1_0 \frac{\ln x}{(1 + x) \sqrt{1 + x^2}} \, dx + \int^\infty_1 \frac{\ln x}{(1 + x) \sqrt{1 + x^2}} \, dx,$$
letting $x \mapsto 1/x$ in the second of these integrals we see that
$$\int^\infty_1 \frac{\ln x}{(1 + x) \sqrt{1 + x^2}} \, dx = -\int^1_0 \frac{\ln x}{(1 + x) \sqrt{1 + x^2}} \, dx.$$
Thus $I_1 = 0$.
For the second of your integrals, we begin again by writing
$$I_2 = \int^1_0 \frac{x^2 \ln x}{(1 + x)(1 + x^2) \sqrt{1 + x^2}} \, dx + \int^\infty_1 \frac{x^2 \ln x}{(1 + x)(1 + x^2) \sqrt{1 + x^2}} \, dx.$$
In the second of these integrals if we again let $x \mapsto 1/x$, then
\begin{align*}
I_2 &= \int^1_0 \frac{x^2 \ln x}{(1 + x)(1 + x^2) \sqrt{1 + x^2}} \, dx - \int^1_0 \frac{\ln x}{(1 + x)(1 + x^2) \sqrt{1 + x^2}} \, dx\\
&= \int^1_0 \frac{(x^2 - 1) \ln x}{(1 + x)(1 + x^2) \sqrt{1 + x^2}} \, dx = \int^1_0 \frac{(x - 1) \ln x}{(1 + x^2)^{3/2}} \, dx.
\end{align*}
Surprisingly (perhaps to some), this last integral can be found in elementary terms. To find it we write
$$\int \frac{(x - 1) \ln x}{(1 + x^2) \sqrt{1 + x^2}} = \int \frac{x \ln x}{(1 + x^2)^{3/2}} \, dx - \int \frac{\ln x}{(1 + x^2)^{3/2}} \, dx = I_\alpha - I_\beta.$$
For the first integral $I_\alpha$, integrating by parts we have
\begin{align*}
I_\alpha &= \int \frac{x}{(1 + x^2)^{3/2}} \cdot \ln x \, dx\\
&= -\frac{\ln x}{\sqrt{1 + x^2}} + \int \frac{dx}{x \sqrt{1 + x^2}} \, dx\\&= -\frac{\ln x}{\sqrt{1 + x^2}} - \ln (\sqrt{1 + x^2} + 1) + \ln x + C_\alpha.
\end{align*}
(The integral that results after applying integration by parts can be found by applying a substitution of $x = \tan \theta$, for example) 
For the second integral $I_\beta$, integrating by parts again gives
\begin{align*}
I_\beta &= \int \frac{1}{(1 + x^2)^{3/2}} \cdot \ln x \, dx\\
&= \frac{x \ln x}{\sqrt{1 + x^2}} - \int \frac{dx}{\sqrt{1 + x^2}}\\
&= \frac{x \ln x}{\sqrt{1 + x^2}} - \sinh^{-1} x + C_\beta.
\end{align*}
So
$$\int \frac{(x - 1) \ln x}{(1 + x^2) \sqrt{1 + x^2}} \, dx = -\frac{(x + 1)\ln x}{\sqrt{1 + x^2}} - \ln (\sqrt{1 + x^2} + 1) + \ln x + \sinh^{-1} x + C.$$
Now, on applying the limits of integration we have
\begin{align*}
I_2 &= \int^1_0 \frac{(x - 1) \ln x}{(1 + x^2) \sqrt{1 + x^2}} \, dx = -\ln (1 + \sqrt{2}) + \ln 2 + \sinh^{-1} (1)\\ &= \ln \left (\frac{2}{1 + \sqrt{2}} \right ) + \ln (1 + \sqrt{2}) = \ln 2,
\end{align*}
giving
$$I = I_1 - I_2 = 0 - \ln 2 = -\ln 2,$$
as expected.
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