Find the in tegration1

$\begin{aligned} I & = \int_0^1 \frac {x\ln(x+1)}{x^2+1}dx & \small \color{#3D99F6} \text{Let }x = \tan \theta \implies dx = \sec^2 \theta \\ & = \int_0^\frac \pi 4 \tan \theta \ln (\tan \theta +1)\ d \theta \\ & = \int_0^\frac \pi 4 \tan \theta \ln \left(\frac {\cos \theta + \sin \theta}{\cos \theta} \right) d \theta \\ & = \int_0^\frac \pi 4 \frac {\sin \theta \ln \left(\sqrt 2\cos \left(\theta - \frac \pi 4\right)\right)}{\cos \theta} \ d \theta - \int_0^\frac \pi 4 \frac {\sin \theta \ln \left(\cos \theta\right)}{\cos \theta} \ d \theta \end{aligned}$