What is the integral of (2x^2+x)/(x^4-2x^2+1) ?

The integral of (2x^2+x)/(x^4-2x^2+1) is

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integral of (2x^2+x)/(x^4-2x^2+1)

\int \frac{2 x ^{2} + x}{x ^{4} - 2 x ^{2} + 1} d x

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

Solution steps

\int \frac{2 x ^{2} + x}{x ^{4} - 2 x ^{2} + 1} d x

Expand

\frac{2 x ^{2} + x}{x ^{4} - 2 x ^{2} + 1} : \frac{2 x ^{2}}{x ^{4} - 2 x ^{2} + 1} + \frac{x}{x ^{4} - 2 x ^{2} + 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{2 x ^{2}}{x ^{4} - 2 x ^{2} + 1} d x + \int \frac{x}{x ^{4} - 2 x ^{2} + 1} d x

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

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

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

Add a constant to the solution

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

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