What is the integral of (2x+1)/(x^6+18x^4+81x^2) ?

The integral of (2x+1)/(x^6+18x^4+81x^2) is

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integral of (2x+1)/(x^6+18x^4+81x^2)

\int \frac{2 x + 1}{x ^{6} + 18 x ^{4} + 81 x ^{2}} d x

\frac{2}{81} ln ∣ x ∣ - \frac{1}{81} ln x^{2} + 9 - \frac{1}{162} arctan (\frac{x}{3}) - \frac{1}{972} sin (2 arctan (\frac{x}{3})) - \frac{1}{81 x} + \frac{1}{9 ( x ^{2} + 9 )} + C

Solution steps

\int \frac{2 x + 1}{x ^{6} + 18 x ^{4} + 81 x ^{2}} d x

Take the partial fraction of

\frac{2 x + 1}{x ^{6} + 18 x ^{4} + 81 x ^{2}} : \frac{2}{81 x} + \frac{1}{81 x ^{2}} + \frac{- 2 x - 1}{81 ( x ^{2} + 9 )} + \frac{- 2 x - 1}{9 ( x ^{2} + 9 ) ^{2}}

= \int \frac{2}{81 x} + \frac{1}{81 x ^{2}} + \frac{- 2 x - 1}{81 ( x ^{2} + 9 )} + \frac{- 2 x - 1}{9 ( x ^{2} + 9 ) ^{2}} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{2}{81 x} d x + \int \frac{1}{81 x ^{2}} d x + \int \frac{- 2 x - 1}{81 ( x ^{2} + 9 )} d x + \int \frac{- 2 x - 1}{9 ( x ^{2} + 9 ) ^{2}} d x

\int \frac{2}{81 x} d x = \frac{2}{81} ln ∣ x ∣

\int \frac{1}{81 x ^{2}} d x = - \frac{1}{81 x}

\int \frac{- 2 x - 1}{81 ( x ^{2} + 9 )} d x = \frac{1}{81} (- ln x^{2} + 9 - \frac{1}{3} arctan (\frac{x}{3}))

\int \frac{- 2 x - 1}{9 ( x ^{2} + 9 ) ^{2}} d x = \frac{1}{9} (\frac{1}{x ^{2} + 9} - \frac{1}{54} (arctan (\frac{x}{3}) + \frac{1}{2} sin (2 arctan (\frac{x}{3}))))

= \frac{2}{81} ln ∣ x ∣ - \frac{1}{81 x} + \frac{1}{81} (- ln x^{2} + 9 - \frac{1}{3} arctan (\frac{x}{3})) + \frac{1}{9} (\frac{1}{x ^{2} + 9} - \frac{1}{54} (arctan (\frac{x}{3}) + \frac{1}{2} sin (2 arctan (\frac{x}{3}))))

Simplify

\frac{2}{81} ln ∣ x ∣ - \frac{1}{81 x} + \frac{1}{81} (- ln x^{2} + 9 - \frac{1}{3} arctan (\frac{x}{3})) + \frac{1}{9} (\frac{1}{x ^{2} + 9} - \frac{1}{54} (arctan (\frac{x}{3}) + \frac{1}{2} sin (2 arctan (\frac{x}{3})))) : \frac{2}{81} ln ∣ x ∣ - \frac{1}{81} ln x^{2} + 9 - \frac{1}{162} arctan (\frac{x}{3}) - \frac{1}{972} sin (2 arctan (\frac{x}{3})) - \frac{1}{81 x} + \frac{1}{9 ( x ^{2} + 9 )}

= \frac{2}{81} ln ∣ x ∣ - \frac{1}{81} ln x^{2} + 9 - \frac{1}{162} arctan (\frac{x}{3}) - \frac{1}{972} sin (2 arctan (\frac{x}{3})) - \frac{1}{81 x} + \frac{1}{9 ( x ^{2} + 9 )}

Add a constant to the solution

= \frac{2}{81} ln ∣ x ∣ - \frac{1}{81} ln x^{2} + 9 - \frac{1}{162} arctan (\frac{x}{3}) - \frac{1}{972} sin (2 arctan (\frac{x}{3})) - \frac{1}{81 x} + \frac{1}{9 ( x ^{2} + 9 )} + C

What is the integral of (2x+1)/(x^6+18x^4+81x^2) ?
The integral of (2x+1)/(x^6+18x^4+81x^2) is