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Differentiate w.r.t. x
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\int \frac{9\left(x-1\right)\left(x+1\right)\left(x^{2}+1\right)}{\left(x-1\right)\left(x+1\right)}\mathrm{d}x
Factor the expressions that are not already factored in \frac{9\left(x^{4}-1\right)}{x^{2}-1}.
\int 9\left(x^{2}+1\right)\mathrm{d}x
Cancel out \left(x-1\right)\left(x+1\right) in both numerator and denominator.
\int 9x^{2}+9\mathrm{d}x
Expand the expression.
\int 9x^{2}\mathrm{d}x+\int 9\mathrm{d}x
Integrate the sum term by term.
9\int x^{2}\mathrm{d}x+\int 9\mathrm{d}x
Factor out the constant in each of the terms.
3x^{3}+\int 9\mathrm{d}x
Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{2}\mathrm{d}x with \frac{x^{3}}{3}. Multiply 9 times \frac{x^{3}}{3}.
3x^{3}+9x
Find the integral of 9 using the table of common integrals rule \int a\mathrm{d}x=ax.
3x^{3}+9x+С
If F\left(x\right) is an antiderivative of f\left(x\right), then the set of all antiderivatives of f\left(x\right) is given by F\left(x\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.