Integral Integral13

$\begin{aligned} I & = \int_0^1 \log \left(\frac 1{x^5} + \frac 1{x^4} + \frac 1{x^3} + \frac 1{x^2} + \frac 1x + 1\right) dx \\ & = \int_0^1 \log \left(\frac {1+x + x^2 + x^3 + x^4 + x^5}{x^5}\right) dx \\ & = \int_0^1 \log \left(\frac {1- x^6}{x^5(1-x)}\right) dx \\ & = {\color{#3D99F6}\int_0^1 \log(1- x^6) \ dx} - 5 \int_0^1 \log x \ dx - \int_0^1 \log (1-x) \ dx & \small \color{#3D99F6} \text{By integration by parts} \\ & = {\color{#3D99F6} x \log (1-x^6) \bigg|_0^1 + \int_0^1 \frac {6x^6}{1- x^6} dx} - 5 (x\log x- x)\bigg|_0^1 - (1-x) \log(1-x) - x \bigg|_0^1 \\ & = {\color{#3D99F6} x \log (1-x^6) \bigg|_0^1 - 6\int_0^1 dx + 6\int_0^1 \frac 1{1- x^6} dx} + 6 \\ & = {\color{#3D99F6} x \log (1-x^6) \bigg|_0^1 - 6 - 6\int_0^1\frac 1{(x-1)(x^2+x+1)(1+x)(x^2-x+1)} dx} + 6 \\ & = x \log (1-x^6) \bigg|_0^1 - 6\int_0^1 \frac 16 \left(\frac 1{x-1} - \frac 1{x+1} + \frac {x-2}{x^2-x+1} - \frac {x+2}{x^2+x+1} \right) dx \\ & = {\color{#3D99F6}x \log (1-x^6) \bigg|_0^1 - \log(x-1)\bigg|_0^1} + \log(x+1)\bigg|_0^1 - \frac 12 \int_0^1 \frac {2x-1-3}{x^2-x+1} dx + \frac 12 \int_0^1 \frac {2x+1+3}{x^2+x+1} dx \\ & = {\color{#3D99F6} \lim_{x \to 1} \left(x\log (1-x^6) - \log (1-x) \right)} + \log 2 + \frac 12 \log \left(\frac {x^2+x+1}{x^2-x+1}\right) \bigg|_0^1 + \frac 32 \int_0^1 \frac 1{x^2-x+1} dx + \frac 32 \int_0^1 \frac 1{x^2+x+1} dx \\ & = {\color{#3D99F6} \lim_{x \to 1} \log \left(\frac {1-x^6}{1-x}\right)} + \log 2 + \frac 12 \log 3 + 2 \int_0^1 \frac 1{\frac 43\left(x-\frac 12\right)^2+1} dx + 2 \int_0^1 \frac 1{\frac 43\left(x+\frac 12\right)^2+1} dx \\ & = {\color{#3D99F6} \lim_{x \to 1} \log \left(\sum_{n=0}^5 x^n\right)} + \log 2 + \frac 12 \log 3 + \sqrt 3 \int_{-\frac 1{\sqrt 3}}^{\frac 1{\sqrt 3}} \frac 1{x^2+1} dx +\sqrt 3 \int_{\frac 1{\sqrt 3}}^{\sqrt 3} \frac 1{x^2+1} dx \\ & = {\color{#3D99F6} \log 6} + \log 2 + \frac 12 \log 3 + \sqrt 3 \int_{-\frac 1{\sqrt 3}}^{\sqrt 3} \frac 1{x^2+1} dx \\ & = \frac 12 \left(2\log 6 + 2 \log 2 + \log 3\right) + \sqrt 3 \tan^{-1} x\ \bigg|_{-\frac 1{\sqrt 3}}^{\sqrt 3} \\ & = \frac 12 \left(\log \left(6^2 \cdot 2^2 \cdot 3\right) \right) + \sqrt 3 \left(\tan^{-1} \sqrt 3 + \tan^{-1} \frac 1{\sqrt 3}\right) \\ & = \frac 12 \log 432 + \sqrt 3 \left(\tan^{-1} \frac {\sqrt 3 + \frac 1{\sqrt 3}}{1 - \sqrt 3 \times \frac 1{\sqrt 3}} \right) \\ & = \frac 12 \log 432 + \sqrt 3 \times \frac \pi 2 \\ & = \frac 12 (\pi \sqrt 3 + \log 432) \end{aligned}$

Therefore, $a+b = 3+432 = \boxed{435}$

$a=3, b=432$

See this problem .