What is the integral of (x^2+2x+2)/(x^3-1) ?

The integral of (x^2+2x+2)/(x^3-1) is

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integral of (x^2+2x+2)/(x^3-1)

\int \frac{x ^{2} + 2 x + 2}{x ^{3} - 1} d x

\frac{1}{3} ln x^{3} - 1 + \frac{4}{3} ln ∣ x - 1 ∣ - \frac{1}{3} ln 4 x^{2} + 4 x + 4 - \frac{2}{3} arctan (\frac{1}{3} (2 x + 1)) + \frac{1}{3} (- ln \frac{4 x ^{2} + 4 x}{3} + \frac{4}{3} + 23 arctan (\frac{2 x + 1}{3})) + C

Solution steps

\int \frac{x ^{2} + 2 x + 2}{x ^{3} - 1} d x

Expand

\frac{x ^{2} + 2 x + 2}{x ^{3} - 1} : \frac{x ^{2}}{x ^{3} - 1} + \frac{2 x}{x ^{3} - 1} + \frac{2}{x ^{3} - 1}

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{x ^{2}}{x ^{3} - 1} d x + \int \frac{2 x}{x ^{3} - 1} d x + \int \frac{2}{x ^{3} - 1} d x

\int \frac{x ^{2}}{x ^{3} - 1} d x = \frac{1}{3} ln x^{3} - 1

\int \frac{2 x}{x ^{3} - 1} d x = 2 (\frac{1}{3} ln ∣ x - 1 ∣ + \frac{1}{6} (- ln \frac{4}{3} + \frac{4 x ^{2} + 4 x}{3} + 23 arctan (\frac{2 x + 1}{3})))

\int \frac{2}{x ^{3} - 1} d x = 2 (\frac{1}{3} ln ∣ x - 1 ∣ - \frac{1}{6} ln 4 x^{2} + 4 x + 4 - \frac{1}{3} arctan (\frac{1}{3} (2 x + 1)))

= \frac{1}{3} ln x^{3} - 1 + 2 (\frac{1}{3} ln ∣ x - 1 ∣ + \frac{1}{6} (- ln \frac{4}{3} + \frac{4 x ^{2} + 4 x}{3} + 23 arctan (\frac{2 x + 1}{3}))) + 2 (\frac{1}{3} ln ∣ x - 1 ∣ - \frac{1}{6} ln 4 x^{2} + 4 x + 4 - \frac{1}{3} arctan (\frac{1}{3} (2 x + 1)))

Simplify

\frac{1}{3} ln x^{3} - 1 + 2 (\frac{1}{3} ln ∣ x - 1 ∣ + \frac{1}{6} (- ln \frac{4}{3} + \frac{4 x ^{2} + 4 x}{3} + 23 arctan (\frac{2 x + 1}{3}))) + 2 (\frac{1}{3} ln ∣ x - 1 ∣ - \frac{1}{6} ln 4 x^{2} + 4 x + 4 - \frac{1}{3} arctan (\frac{1}{3} (2 x + 1))) : \frac{1}{3} ln x^{3} - 1 + \frac{4}{3} ln ∣ x - 1 ∣ - \frac{1}{3} ln 4 x^{2} + 4 x + 4 - \frac{2}{3} arctan (\frac{1}{3} (2 x + 1)) + \frac{1}{3} (- ln \frac{4 x ^{2} + 4 x}{3} + \frac{4}{3} + 23 arctan (\frac{2 x + 1}{3}))

= \frac{1}{3} ln x^{3} - 1 + \frac{4}{3} ln ∣ x - 1 ∣ - \frac{1}{3} ln 4 x^{2} + 4 x + 4 - \frac{2}{3} arctan (\frac{1}{3} (2 x + 1)) + \frac{1}{3} (- ln \frac{4 x ^{2} + 4 x}{3} + \frac{4}{3} + 23 arctan (\frac{2 x + 1}{3}))

Add a constant to the solution

= \frac{1}{3} ln x^{3} - 1 + \frac{4}{3} ln ∣ x - 1 ∣ - \frac{1}{3} ln 4 x^{2} + 4 x + 4 - \frac{2}{3} arctan (\frac{1}{3} (2 x + 1)) + \frac{1}{3} (- ln \frac{4 x ^{2} + 4 x}{3} + \frac{4}{3} + 23 arctan (\frac{2 x + 1}{3})) + C

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What is the integral of (x^2+2x+2)/(x^3-1) ?
The integral of (x^2+2x+2)/(x^3-1) is