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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

\int \frac{2}{5 x} d x

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\int sec (x) (sec (x) tan (x)) d x

\int \frac{tan ( x )}{cos ( x ) sec ( sec ( x ) )} d x

\int \frac{x - 2}{3 x ^{2} - 4 x + 6} d x

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