Analytic form of: $ \int \frac{\bigl[\cos^{-1}(x)\sqrt{1-x^2}\bigr]^{-1}}{\ln\bigl( 1+\sin(2x\sqrt{1-x^2})/\pi\bigr)} dx $

Let $$J(x)=\int\dfrac1{\ln\left(1+\dfrac1\pi\sin\left(2x\sqrt{1-x^2}\right)\right)}\dfrac{\mathrm dx}{\sqrt{1-x^2}\cos^{-1}x}.\tag1$$ Firstly, $$1-\left(2x\sqrt{1-x^2}\right)^2 = 1-4x^2+4x^4 = (2x^2-1)^2 = (\cos(2\cos^{-1}x))^2,$$ so $$J(x)=\int\dfrac{d(\cos^{-1}x)}{\cos^{-1}x\cdot\ln\left(1+\dfrac1\pi\cos\left(\cos(2\cos^{-1}x)\right)\right)}=J_1(\cos^{-1}x),\tag2$$ where $$J_1(y)=\int\dfrac{\mathrm dy}{y\ln\left(1+\dfrac1\pi\cos(\cos 2y)\right)}.\tag3$$ I have not obtained closed form for $(3).$

At the same time, it is possible to obtain Taylor series of $${\small \dfrac1{\ln\left(1+\dfrac1\pi\cos(z)\right)} = \dfrac1p\left(1+\dfrac1{2q}y^2 + \dfrac{6+(2-\pi)p}{24q^2}y^4+\dfrac{90+30(2-\pi)p+(16-13\pi+\pi^2)p^2}{720q^3}y^6+\dots\right)},$$ where $$p=\ln\left(1+\dfrac1\pi\right),\quad q=(\pi+1)\ln\left(1+\dfrac1\pi\right)\tag4$$ (see also Wolfram Alpha), and this approximation leads to the formula of $$\begin{align} &J_1(y)\approx\dfrac1{23040(1+\pi)^3p^4}\Big( \left(90+(60-30\pi)p+(\pi^2-13\pi+16)p^2\right)\mathrm{Ci}(12y)\\ &-6\left(-90 - (90\pi+180)p + (19\pi^2-7\pi -56)p^2\right)\mathrm{Ci}(8y)\\ &+15\left(90+(252+162\pi)p + (464+787\pi+353\pi^2)p^2\right)\mathrm{Ci}(4y)\\ &+\big((23040\pi^3+69120\pi^2+69120\pi+23040)p^3+(5410\pi^2+11750\pi+6640)p^2\\ &+(1860\pi+2760)p+900\big)\ln(2y)+\dots\Big) \end{align}$$ (see also Wolfram Alpha).