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

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

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

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

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

Solution steps

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

Take the partial fraction of

\frac{x + 2}{( x ^{2} - 1 ) ( x ^{2} + 1 ) ^{2}} : - \frac{1}{8 ( x + 1 )} + \frac{3}{8 ( x - 1 )} + \frac{- x - 2}{4 ( x ^{2} + 1 )} + \frac{- x - 2}{2 ( x ^{2} + 1 ) ^{2}}

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

\int \frac{1}{8 ( x + 1 )} d x = \frac{1}{8} ln ∣ x + 1 ∣

\int \frac{3}{8 ( x - 1 )} d x = \frac{3}{8} ln ∣ x - 1 ∣

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

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

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

Add a constant to the solution

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

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