Log Trig Integral 3

$\begin{aligned} I & = \int_0^\frac \pi 4 \log(\cot x + 1) dx \\ & = \int_0^\frac \pi 4 \log \left(\frac {\cos x + \sin x}{\sin x} \right) dx \\ & = \int_0^\frac \pi 4 \log \left(\frac {\sqrt 2 \sin \left(x + \frac \pi 4\right)}{\sin x} \right) dx \\ & = \int_0^\frac \pi 4 \frac 12 \log 2 \ dx + {\color{#3D99F6}\int_0^\frac \pi 4 \log \left(\sin \left(x + \frac \pi 4\right) \right) dx} - \int_0^\frac \pi 4 \log \left(\sin x \right) dx & \small \color{#3D99F6} \text{By }\int_a^b f(x) \ dx = \int_a^b f(a+b-x) \ dx \\ & = \frac {\pi \log 2}8 + {\color{#3D99F6}\int_0^\frac \pi 4 \log \left(\sin \left(\frac \pi 2 - x \right) \right) dx} - {\color{#D61F06}\int_0^\frac \pi 4 \log \left(2\sin x \right) dx} + \int_0^\frac \pi 4 \log 2 \ dx & \small \color{#D61F06} \text{See reference: }G = - 2 \int_0^\frac \pi 4 \log \left(2\sin x \right) dx \\ & = \frac {\pi \log 2}8 + {\color{#3D99F6}\int_0^\frac \pi 4 \log \left(\cos x \right) dx} + {\color{#D61F06} \frac G2} + \color{#3D99F6} \int_0^\frac \pi 4 \log 2 \ dx \\ & = \frac {\pi \log 2}8 + {\color{#3D99F6}\int_0^\frac \pi 4 \log \left(2\cos x \right) dx} + \frac G2 & \small \color{#3D99F6} \text{See reference: }G = 2 \int_0^\frac \pi 4 \log \left(2\cos x \right) dx \\ & = \frac {\pi \log 2}8 + {\color{#3D99F6}\frac G2} + \frac G2 \\ & = \frac {\pi \log 2}8 + G \end{aligned}$

$\implies a = \boxed{8}$

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Reference: $G$ , the Catalan's constant