How to evaluate $\int \frac{\sqrt{1+x}+\sqrt{1-x}}{\sqrt{1+x}-\sqrt{1-x}} dx$? I am trying to evaluate
$$\int \frac{\sqrt{1+x}+\sqrt{1-x}}{\sqrt{1+x}-\sqrt{1-x}} dx \quad (1)$$

The typical way to confront this kind of integrals are the conjugates i.e:
$$\int \frac{\sqrt{1+x}+\sqrt{1-x}}{\sqrt{1+x}-\sqrt{1-x}} dx  = $$
$$ \int \left(\frac{\sqrt{1+x}+\sqrt{1-x}}{\sqrt{1+x}-\sqrt{1-x}}\right)\left(\frac{\sqrt{1+x}+\sqrt{1-x}}{\sqrt{1+x}+\sqrt{1-x}}\right)\left(\frac{\sqrt{1+x}-\sqrt{1-x}}{\sqrt{1+x}-\sqrt{1-x}}\right) dx = $$
$$\int \left(\frac{(\sqrt{1+x})^2-(\sqrt{1-x})^2}{(\sqrt{1+x})^2-(\sqrt{1-x})^2}\right)\left(\frac{\sqrt{1+x}-\sqrt{1-x}}{\sqrt{1+x}+\sqrt{1-x}}\right)dx = $$
$$\int 1*\left(\frac{\sqrt{1+x}-\sqrt{1-x}}{\sqrt{1+x}+\sqrt{1-x}}\right)dx $$
That's a dead end.

I also tried other conjugate approaches (only the numerator, only the denominator etc) with no better luck.
Any ideas?

Multiplying by the conjugate is not a dead end.  I'm not sure why you multiplied by the conjugate of the numerator and denominator, you can easily evaluate this integral just by multiplying by the conjugate of the denominator:
$$I=\int \left(\frac{\sqrt{1+x}+\sqrt{1-x}}{\sqrt{1+x}-\sqrt{1-x}}\right)\left(\frac{\sqrt{1+x}+\sqrt{1-x}}{\sqrt{1+x}+\sqrt{1-x}}\right) dx$$
$$I=\int \frac{{(1+x)}+(1-x)+2\sqrt{1-x^2}}{2x} dx$$
$$I=\ln{|x|}+\int \frac{\sqrt{1-x^2}}{x} \; dx$$
Let $x=\sin{\theta}$ to evaluate the integral:
$$I=\ln{|x|}+\int \frac{\cos^2{\theta}}{\sin{\theta}} \; d\theta $$
$$I=\ln{|x|}+\int \csc{\theta} \; d\theta - \int \sin{\theta} \; d\theta$$
$$I=\ln{|x|}-\ln{\big | \csc{\theta}+\cot{\theta}\big |}+ \cos{\theta}+C$$
Substitute $\theta$ back for $x$:
$$I=2\ln{|x|}-\ln{\big | \sqrt{1-x^2}+1\big |} + \sqrt{1-x^2}+C$$
