$$
\begin{align}
\int_0^1\log(1+x)\frac{\mathrm{d}x}{1+x^2}
&=\int_0^1\log\left(\frac2{1+x}\right)\frac{\mathrm{d}x}{1+x^2}\\
&=\frac\pi8\log(2)\tag{1}
\end{align}
$$
$$
f(a)=\int_0^1x\log(a+x)\frac{\mathrm{d}x}{1+x^2}\tag{2}
$$
$$
\begin{align}
f(0)
&=\int_0^1x\log(x)\frac{\mathrm{d}x}{1+x^2}\\
&=\frac12\int_0^1\log(x)\mathrm{d}\log(1+x^2)\\
&=-\frac12\int_0^1\frac{\log(1+x^2)}{x}\mathrm{d}x\\
&=-\frac{\pi^2}{48}\tag{3}
\end{align}
$$
$$
\begin{align}
f'(a)
&=\int_0^1\left[\frac{ax+1}{1+x^2}-\frac{a}{a+x}\right]\frac{\mathrm{d}x}{1+a^2}\\
Question:
if the postive integer $n$ such $$w(n)+\varphi{(n)}-\pi{(n)}=1$$
find the $\max{(n)}$,
where $\varphi{(n)}$ is Euler's totient function (Euler function)
$\pi{(n)}$ is prime-counting function (prime-counting function)
$w(n)$ is represents the number of distinct prime fact...
I ran across these two notations for the log function (squared), which one is more conventional.
$\log^2(n)$ or $[\log(n)]^2$
