Evaluating the indefinite integral $\int\log\!\left(x+\sqrt{x^2-1}\right)\!dx$ I came across the following integral, and I don't know how to solve it.
$$
\int\log\left(x+\sqrt{x^2-1}\right)dx
$$
I tried the "obvious" substitution of $x=\sec\theta$, which gives you:
$$
\int\tan\theta\sec\theta\log\left(\tan\theta+\sec\theta\right)d\theta
$$
However, this doesn't simplify it much, in the sense that I have no idea how to solve this now! Take out the common factor of $\sec$ from the log, or perhaps do $u=\tan\theta+\sec\theta$, but both of these lead to dead ends (at least for me).
In case you are wondering, Wolfram|Alpha claims the answer is:
$$x\log\left(x+\sqrt{x^2-1}\right)-\sqrt{x^2-1}+c$$
 A: Computing directly shows that the integrand, $$\log\!\left(x + \sqrt{x^2 - 1}\right),$$ is the inverse of (the restriction to $[0, \infty)$ of) the hyperbolic cosine function $$\cosh u := \frac{e^u + e^{-u}}{2};$$ because of this, the integrand here is usually denoted $$\text{arcosh x}.$$
This suggests that we may proceed analogously to the usual derivation of the antiderivatives of inverse trigonometric functions: Substituting $x = \cosh u$ gives
$$\int \text{arcosh } x \,dx = \int \text{arcosh} (\cosh u) \,d(\cosh u) = \int u \sinh u \,du.$$
Applying integration by parts with $v = u$, $dw = \sinh u \,du$ gives that this is
$$u \cosh u - \int \cosh u\,du = u \cosh u - \sinh u + C,$$ and reverse-substituting to write this in terms of $x$ yields
$$\text{arcosh } x \cosh (\text{arcosh } x) - \sinh(\text{arcosh } x) + C.$$
Substituting $u = \text{arcosh x}$ in the familiar identity $$\cosh^2 u = \sinh^2 u + 1,$$ simplifying, rearranging, and using that $\text{arcosh}$ is nonnegative (or, alternatively, appealing to the hyperbolic analogue of a reference triangle) gives the identity
$$\sinh (\text{arcosh } x) = \sqrt{x^2 - 1}.$$ Substituing in the above expression gives the antiderivative,
$$\color{#bf0000}{\int \text{arcosh } x \,dx = x \,\text{arcosh } x - \sqrt{x^2 - 1} + C},$$ which in particular agrees with the result given by WolframAlpha.
A: Integrate by parts:
$$
x\log(x+\sqrt{x^2-1})-\int x\frac{1+\dfrac{x}{\sqrt{x^2-1}}}{x+\sqrt{x^2-1}}\,dx
$$

Now, where did I see $\log(x+\sqrt{x^2-1})$ again? Set $x+\sqrt{x^2-1}=e^t$, so
$$
x^2-1=e^{2t}-2xe^t+x^2
$$
or
$$
2xe^t=e^{2t}+1
$$
and
$$
x=\cosh t
$$
So a good substitution could be this one, wouldn't it? The integral becomes
$$
\int t\sinh t\,dt=t\cosh t-\int \cosh t\,dt=t\cosh t-\sinh t
$$
and now it's just back substitution.
A: Recognize $\log\left(x+\sqrt{x^2-1}\right)=\cosh^{-1}x$ and then integrate by parts
$$
\int\log\left(x+\sqrt{x^2-1}\right)dx=\int \cosh^{-1}x\>dx
= x \cosh^{-1}x-\int \frac x{\sqrt{x^2-1}}dx
$$
where the remaining integral is just $\sqrt{x^2-1}$.
A: $\displaystyle \int ln(x+\sqrt{x^{2}-1})dx $
$\displaystyle {we've got
}: \; 
\fbox{ 
$ ch^{2}(y)-sh^{2}(y)=1,ch(y)=\frac{e^{y}+e^{-y}}{2},sh(y)=\frac{e^{y}-e^{-y}}{2}$
}\\$
$\displaystyle  x=ch(y)\Rightarrow dx=sh(y)dy$
$\displaystyle   ln(x+\sqrt{x^{2}-1})=ln(ch(y)+sh(y))=y$
$\displaystyle \int ln(x+\sqrt{x^{2}-1})dx =\int y \ sh({y})dy=\int ych'(y)dy$
${we've got
}: \;\fbox{ 
$ \displaystyle \int fg'=fg-\int f'g $
}$
$\displaystyle \int y \ sh({y})dy=y \ ch({y})-\int ch(y)dy=y \ ch({y})-\ sh({y})+c$
$\displaystyle \int ln(x+\sqrt{x^{2}-1})dx=xch^{-1}(x) -\sqrt{x^{2}-1}+c$
$\displaystyle =xln(x+\sqrt{x^{2}-1})-\sqrt{x^{2}-1}+c$
