What is the integral of (x-2)/((x^2+8x+64)^2) ?

The integral of (x-2)/((x^2+8x+64)^2) is

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integral of (x-2)/((x^2+8x+64)^2)

\int \frac{x - 2}{( x ^{2} + 8 x + 64 ) ^{2}} d x

\frac{1}{768 3} (2 arctan (\frac{1}{4 3} x + \frac{1}{3}) + sin (2 arctan (\frac{1}{4 3} x + \frac{1}{3}))) (x - 2) - \frac{1}{384 3} arctan (\frac{x + 4}{4 3}) (x + 4) + C

Solution steps

\int \frac{x - 2}{( x ^{2} + 8 x + 64 ) ^{2}} d x

Apply Integration By Parts

= \frac{1}{768 3} (2 arctan (\frac{1}{4 3} x + \frac{1}{3}) + sin (2 arctan (\frac{1}{4 3} x + \frac{1}{3}))) (x - 2) - \int \frac{1}{768 3} (2 arctan (\frac{1}{4 3} x + \frac{1}{3}) + sin (2 arctan (\frac{1}{4 3} x + \frac{1}{3}))) d x

\int \frac{1}{768 3} (2 arctan (\frac{1}{4 3} x + \frac{1}{3}) + sin (2 arctan (\frac{1}{4 3} x + \frac{1}{3}))) d x = \frac{1}{384 3} arctan (\frac{x + 4}{4 3}) (x + 4)

= \frac{1}{768 3} (2 arctan (\frac{1}{4 3} x + \frac{1}{3}) + sin (2 arctan (\frac{1}{4 3} x + \frac{1}{3}))) (x - 2) - \frac{1}{384 3} arctan (\frac{x + 4}{4 3}) (x + 4)

Add a constant to the solution

= \frac{1}{768 3} (2 arctan (\frac{1}{4 3} x + \frac{1}{3}) + sin (2 arctan (\frac{1}{4 3} x + \frac{1}{3}))) (x - 2) - \frac{1}{384 3} arctan (\frac{x + 4}{4 3}) (x + 4) + C

What is the integral of (x-2)/((x^2+8x+64)^2) ?
The integral of (x-2)/((x^2+8x+64)^2) is