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

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

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

integral

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

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

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Apply Integration By Parts

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

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

= - 4 (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 )})

Simplify

- 4 (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 )}) : - x ln ∣ x + 1 ∣ + \frac{x}{x + 1} + x ln ∣ x - 1 ∣ + \frac{x}{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 )}{( x + 1 ) ( x - 1 )}

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

Add a constant to the solution

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

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