<s>[INST] Evaluate $\int_{-1}^{1} \cot^{-1} \left(\frac{1}{\sqrt{1-x^2}}\right) \cot^{-1}\left(\raise{3pt}\frac{x}{\sqrt{1-(x^2)^{|x|}}}\right)$ 
$$\int_{-1}^{1} \left(\cot^{-1} \dfrac{1}{\sqrt{1-x^2}}\right) \left(\cot^{-1}\dfrac{x}{\sqrt{1-(x^2)^{|x|}}}\right)= \dfrac{\pi^2(\sqrt a-\sqrt b )}{\sqrt c}$$
  , where a,b, c are natural numbers and are there in their lowest form, then find the value of a+b+c.

Using $\int_a^b f(x)dx= \int_a^bf(a+b-x{) dx}$, I got: 
$$2I =2\pi\int_{0}^1 \cot^{-1}\left(\dfrac{1}{\sqrt{1-x^2}}\right) dx$$
Then, letting $x = \sin \theta$
$I = \pi\displaystyle\int_{0}^{\pi/2}\arctan (\cos\theta) \cos \theta d\theta$
After this I tried integration by parts but it gets really complicated with that? How do I continue? 
EDIT:
Please note that arccot(x) + arccot(-x)= $\pi$ $\ne 0$
Principal range of $\cot^{-1}x$ considered in the question is $(0,\pi)$
 [/INST] I usually see $\cot^{-1}(-x)=-\cot^{-1}(x)$, but working with what is stated in the question,
$$
\begin{align}
&\int_{-1}^1\cot^{-1}\left(\frac1{\sqrt{1-x^2}}\right)\cot^{-1}\left(\raise{3pt}\frac{x}{\sqrt{1-\left(x^2\right)^{|x|}}}\right)\,\mathrm{d}x\tag1\\
&=\int_0^1\cot^{-1}\left(\frac1{\sqrt{1-x^2}}\right)\cot^{-1}\left(\raise{3pt}\frac{x}{\sqrt{1-\left(x^2\right)^{|x|}}}\right)\,\mathrm{d}x\\
&+\int_0^1\cot^{-1}\left(\frac1{\sqrt{1-x^2}}\right)\left[\pi-\cot^{-1}\left(\raise{3pt}\frac{x}{\sqrt{1-\left(x^2\right)^{|x|}}}\right)\right]\,\mathrm{d}x\tag2\\
&=\pi\int_0^1\tan^{-1}\left(\sqrt{1-x^2}\right)\,\mathrm{d}x\tag3\\
&=\pi\int_0^1\frac{\sqrt{1-x^2}}{1+x^2}\,\mathrm{d}x\tag4\\
&=\pi\int_0^{\pi/2}\frac{\cos^2(x)}{1+\sin^2(x)}\,\mathrm{d}x\tag5\\
&=\pi\int_0^{\pi/2}\frac2{1+\sin^2(x)}\,\mathrm{d}x-\frac{\pi^2}2\tag6\\
&=\pi\int_0^{\pi/2}\frac{2\sec^2(x)}{1+2\tan^2(x)}\,\mathrm{d}x-\frac{\pi^2}2\tag7\\
&=\pi\sqrt2\int_0^\infty\frac1{1+u^2}\,\mathrm{d}u-\frac{\pi^2}2\tag8\\
&=\pi^2\frac{\sqrt2-1}2\tag9
\end{align}
$$
Explanation:
$(2)$: use $\cot^{-1}(-x)=\pi-\cot^{-1}(x)$ and substitute $x\mapsto-x$ on $[-1,0]$
$(3)$: $\cot^{-1}(x)=\tan^{-1}\left(\frac1x\right)$
$(4)$: substitute $x\mapsto\sqrt{1-x^2}$ then integrate by parts
$(5)$: substitute $x\mapsto\sin(x)$
$(6)$: add $1$ to the integrand and subtract $\frac{\pi^2}2$
$(7)$: multiply numerator and denominator of the integrand by $\sec^2(x)$
$(8)$: substitute $u=\sqrt2\tan(x)$
$(9)$: substitute $u=\tan(\theta)$ and integrate
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