Limit of $\int_0^1\left(\frac {2}{\sqrt {(1-t^2)(1-xt^2)}}-\frac{x}{1-xt}\right)\,dt$ as $x\to 1^{-}$ While going through this question I was reminded of one of my earlier questions and I found that there is some unfinished business which needs some further exploration.
Let $$F(x) =\int_0^1\left(\frac{2}{\sqrt{(1-t^2)(1-xt^2)}} -\frac{x}{1-xt}\right)\, dt=\int_0^1 f(x, t) \, dt\tag{1}$$ for $x\in[0,1]$.
Let's observe that $$\lim_{x\to 1^-}f(x,t)=\frac{1}{1+t}\tag{2}$$ and hence it is natural to expect that $\lim_{x\to 1^-}F(x)$ should equal $\int_0^1 dt/(1+t)=\log 2$ but numerical evidence as well as some amount of elliptic function theory tells (see one of my questions linked earlier for details) us that this particular limit is $4\log 2$.
This suggests that there is some weird behavior of integrand as $x\to 1^{-}$ (in particular the convergence is not uniform).
I would like to have this limit evaluated using some analysis related to convergence of integrand as $x\to 1^-$. Any help in this direction would be appreciated.

Note: I have asked a new question instead of bumping an old one. The old question is more about solution verification and is related to elliptic integrals. I wanted to have a different perspective which involves general issues of uniform convergence to handle the limit of this integral.
 A: The non-uniform convergence prevents the direct use of the given integrand.
However, the original problem is to compute $L=\lim\limits_{x\to1^-}\big(2K(\sqrt x)+\log(1-x)\big)$.
One may replace $x/(1-xt)$ with "something better" that approximates the original "elliptic" integrand uniformly (on $t\in(0,1)$ as $x\to1^-$) and is still elementarily integrable.
Let's replace $x$ by $x^2$ (to get rid of $\sqrt x$ everywhere), and consider $$F(x,t)=\frac1{\sqrt{(1-t^2)(1-x^2 t^2)}},\\G(x,t)=\frac1{a_x(1+t)\sqrt{(1-t)(1-xt)}},$$ where $a_x=\sqrt{(1+x)/2}$ is chosen to have $\lim\limits_{x\to1^-}\big(F(x,t)-G(x,t)\big)=0$ uniformly.
Now $\int_0^1 F(x,t)\,dt=K(x)$ and $\int_0^1 G(x,t)\,dt=a_x^{-2}\tanh^{-1}a_x$. Hence
\begin{align*}
L&=\lim_{x\to1^-}\big(2K(x)+\log(1-x^2)\big)
\\&=\lim_{x\to1^-}\big(\frac{1}{a_x^2}\log\frac{1+a_x}{1-a_x}+\log(1-x^2)\big)
\\&=\lim_{x\to1^-}\left[\log\left(\frac{1+a_x}{1-a_x}(1-x^2)\right)+\frac{1-x}{1+x}\log\frac{1+a_x}{1-a_x}\right]
\\&=\lim_{x\to1^-}\log\left(\frac{(1+a_x)^2}{1-a_x^2}(1-x^2)\right)
\\&=\lim_{x\to1^-}\log\big(2(1+x)(1+a_x)^2\big)=4\log 2.
\end{align*}
