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\frac{x}{x-1}+\frac{x+1}{\left(x-1\right)\left(x+1\right)}
Factor the expressions that are not already factored in \frac{x+1}{x^{2}-1}.
\frac{x}{x-1}+\frac{1}{x-1}
Cancel out x+1 in both numerator and denominator.
\frac{x+1}{x-1}
Since \frac{x}{x-1} and \frac{1}{x-1} have the same denominator, add them by adding their numerators.
\frac{x}{x-1}+\frac{x+1}{\left(x-1\right)\left(x+1\right)}
Factor the expressions that are not already factored in \frac{x+1}{x^{2}-1}.
\frac{x}{x-1}+\frac{1}{x-1}
Cancel out x+1 in both numerator and denominator.
\frac{x+1}{x-1}
Since \frac{x}{x-1} and \frac{1}{x-1} have the same denominator, add them by adding their numerators.