Rewrite the expression $\frac{x^2}{x^4+8x^2+16}$ inside the integral in factored form
Rewrite the fraction $\frac{x^2}{\left(x^{2}+4\right)^{2}}$ in $2$ simpler fractions using partial fraction decomposition
Expand the integral $\int\left(\frac{1}{x^{2}+4}+\frac{-4}{\left(x^{2}+4\right)^{2}}\right)dx$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately
The integral $\int\frac{1}{x^{2}+4}dx$ results in: $\frac{1}{2}\arctan\left(\frac{x}{2}\right)$
The integral $\int\frac{-4}{\left(x^{2}+4\right)^{2}}dx$ results in: $-\frac{1}{2}\left(\frac{1}{2}\arctan\left(\frac{x}{2}\right)+\frac{x}{x^{2}+4}\right)$
Gather the results of all integrals
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
Expand and simplify
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