Evaluate $\int \frac {\operatorname d\!x} {2x \sqrt{1-x}\sqrt{2-x + \sqrt{1-x}}}$ $$\int \dfrac {\operatorname d\!x} {2x \sqrt{1-x}\sqrt{2-x + \sqrt{1-x}}}$$
Hey there, I've got this complicated integral to evaluate, but I don't know how to go about. I have tried making two substitutions:


*

*$ t^2 = 1 - x $

*$ x = \sin^2\theta $
But both gave another complicated integral to evaluate:
$$ \int \dfrac {\operatorname d\!t} {(t^2-1)\sqrt{ t^2 + t + 1 }} $$
I tried to get the answer for this one using wolfram alpha, but it gave a HUGE, simply HUGE solution. I also tried to get the solution for the original question via wolfram alpha, but it timed out.
Any ideas?
 A: The following integral
$$ \int \frac{1}{x \sqrt{ax^2 + bx + c}}\text{d}x$$
can be solved by substituting $x = \frac{1}{t}$ to get
$$\int \frac{\pm 1}{\sqrt{a + bt + ct^2}}\text{d}t$$
Which can be recast as the derivative of an inverse trigonometric function (could be hyperbolic, depending on the signs taken):
$$\int \frac{\pm 1}{\sqrt{\pm 1 \pm x^2}}$$
And as noted by Ron, your integral is the sum of two such integrals.
A: For what it's worth, you can break things up a bit by recognizing that
$$\frac{1}{t^2-1} = \frac12 \left ( \frac{1}{t-1} - \frac{1}{t+1}\right )$$
For example, consider the $t-1$ piece; you may substitute $u=t+1/2$ and get
$$\int \frac{dt}{(t-1) \sqrt{t^2+t+1}} = \int \frac{du}{(u-3/2) \sqrt{u^2+3/4}}$$
This latter integral is relatively tame according to WA:
$$ \frac{1}{\sqrt{3}} \left[\log{\left(u-\frac{3}{2}\right)}-\log{\left(\sqrt{12 u^2+9}+3 u+\frac{3}{2}\right)}\right]+C$$
where $C$ is a constant of integration.  A similar expression may be found for the $t+1$ piece.
A: The $t$-integral is primed for the well-known Euler substitution. Let
$$u=\frac{\sqrt{t^2+t+1} - 1}t \implies t  = -\frac{1-2u}{1-u^2} \implies dt = \frac{2u^2-2u+2}{(1-u^2)^2} \, du$$
to transform to
$$\begin{align*}
I &= \int \dfrac{dt}{(t^2-1)\sqrt{ t^2 + t + 1 }} \\[1ex]
&= 2 \int \frac{u^2 - u + 1}{(1-u^2)^2 \left(\frac{(1-2u)^2}{(1-u^2)^2} - 1\right) \left(1-\frac{u-2u^2}{1-u^2}\right)} \, du \\[1ex]
&= 2 \int  \frac{u^2-1}{u(u-2)(u^2+2u-2)} \, du
\end{align*}$$
