<s>[INST] Calculate $\int {\frac{{\sqrt {x + 1} - \sqrt {x - 1} }}{{\sqrt {x + 1} + \sqrt {x - 1} }}} dx $ Calculate
$$\int {\frac{{\sqrt {x + 1}  - \sqrt {x - 1} }}{{\sqrt {x + 1}  + \sqrt {x - 1} }}} dx
$$
My try:
$$\int {\frac{{\sqrt {x + 1}  - \sqrt {x - 1} }}{{\sqrt {x + 1}  + \sqrt {x - 1} }}} dx = \left| {x + 1 = {u^2}} \right| = 2\int {\frac{{(u - \sqrt { - 2 + {u^2}} )}}{{u + \sqrt { - 2 + {u^2}} }}du} 
$$
I tried to do first Euler substitute:
$$\sqrt { - 2 + {u^2}}  = {u_1} - u
$$
But it did not lead me to the goal. Any thoughts will be appriciated.
 [/INST] Try 
$$\left (\sqrt{x+1}+\sqrt{x-1} \right )\left (\sqrt{x+1}-\sqrt{x-1} \right ) = 2$$
Then the integral is
$$\frac12 \int dx \, \left (\sqrt{x+1}-\sqrt{x-1} \right )^2$$
which is
$$\int dx \left (x -  \sqrt{x^2-1}\right ) $$
Can you take it from here?
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