What is the integral of 1/((x+1)(x^2+x+1)^2) ?

The integral of 1/((x+1)(x^2+x+1)^2) is

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integral of 1/((x+1)(x^2+x+1)^2)

\int \frac{1}{( x + 1 ) ( x ^{2} + x + 1 ) ^{2}} d x

ln ∣ x + 1 ∣ - \frac{1}{2} ln 4 x^{2} + 4 x + 4 + \frac{1}{3} arctan (\frac{1}{3} (2 x + 1)) + \frac{1}{2 ( x ^{2} + x + 1 )} + \frac{2}{3 3} arctan (\frac{1}{3} (2 x + 1)) + \frac{1}{3 3} sin (2 arctan (\frac{1}{3} (2 x + 1))) + C

Solution steps

\int \frac{1}{( x + 1 ) ( x ^{2} + x + 1 ) ^{2}} d x

Take the partial fraction of

\frac{1}{( x + 1 ) ( x ^{2} + x + 1 ) ^{2}} : \frac{1}{x + 1} - \frac{x}{x ^{2} + x + 1} - \frac{x}{( x ^{2} + x + 1 ) ^{2}}

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

\int \frac{1}{x + 1} d x = ln ∣ x + 1 ∣

\int \frac{x}{x ^{2} + x + 1} d x = \frac{1}{2} ln 4 x^{2} + 4 x + 4 - \frac{1}{3} arctan (\frac{2 x + 1}{3})

\int \frac{x}{( x ^{2} + x + 1 ) ^{2}} d x = 8 (- \frac{1}{4 ( 4 x ^{2} + 4 x + 4 )} - \frac{1}{24 3} (2 arctan (\frac{1}{3} (2 x + 1)) + sin (2 arctan (\frac{1}{3} (2 x + 1)))))

= ln ∣ x + 1 ∣ - (\frac{1}{2} ln 4 x^{2} + 4 x + 4 - \frac{1}{3} arctan (\frac{2 x + 1}{3})) - 8 (- \frac{1}{4 ( 4 x ^{2} + 4 x + 4 )} - \frac{1}{24 3} (2 arctan (\frac{1}{3} (2 x + 1)) + sin (2 arctan (\frac{1}{3} (2 x + 1)))))

Simplify

ln ∣ x + 1 ∣ - (\frac{1}{2} ln 4 x^{2} + 4 x + 4 - \frac{1}{3} arctan (\frac{2 x + 1}{3})) - 8 (- \frac{1}{4 ( 4 x ^{2} + 4 x + 4 )} - \frac{1}{24 3} (2 arctan (\frac{1}{3} (2 x + 1)) + sin (2 arctan (\frac{1}{3} (2 x + 1))))) : ln ∣ x + 1 ∣ - \frac{1}{2} ln 4 x^{2} + 4 x + 4 + \frac{1}{3} arctan (\frac{1}{3} (2 x + 1)) + \frac{1}{2 ( x ^{2} + x + 1 )} + \frac{2}{3 3} arctan (\frac{1}{3} (2 x + 1)) + \frac{1}{3 3} sin (2 arctan (\frac{1}{3} (2 x + 1)))

= ln ∣ x + 1 ∣ - \frac{1}{2} ln 4 x^{2} + 4 x + 4 + \frac{1}{3} arctan (\frac{1}{3} (2 x + 1)) + \frac{1}{2 ( x ^{2} + x + 1 )} + \frac{2}{3 3} arctan (\frac{1}{3} (2 x + 1)) + \frac{1}{3 3} sin (2 arctan (\frac{1}{3} (2 x + 1)))

Add a constant to the solution

= ln ∣ x + 1 ∣ - \frac{1}{2} ln 4 x^{2} + 4 x + 4 + \frac{1}{3} arctan (\frac{1}{3} (2 x + 1)) + \frac{1}{2 ( x ^{2} + x + 1 )} + \frac{2}{3 3} arctan (\frac{1}{3} (2 x + 1)) + \frac{1}{3 3} sin (2 arctan (\frac{1}{3} (2 x + 1))) + C

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What is the integral of 1/((x+1)(x^2+x+1)^2) ?
The integral of 1/((x+1)(x^2+x+1)^2) is