What is the integral of (8-2x)/((x^2-8x+25)^2) ?

The integral of (8-2x)/((x^2-8x+25)^2) is

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integral of (8-2x)/((x^2-8x+25)^2)

\int \frac{8 - 2 x}{( x ^{2} - 8 x + 25 ) ^{2}} d x

\frac{4}{27} (arctan (\frac{1}{3} (x - 4)) + \frac{1}{2} sin (2 arctan (\frac{1}{3} (x - 4)))) - 2 (\frac{1}{108} x (2 arctan (\frac{x - 4}{3}) + sin (2 arctan (\frac{x - 4}{3}))) - \frac{x ^{3} arctan ( \frac{1}{3} ( x - 4 ) ) + 3 x ^{2} - 12 x ^{2} arctan ( \frac{1}{3} ( x - 4 ) ) + 57 x arctan ( \frac{1}{3} ( x - 4 ) ) - 24 x - 100 arctan ( \frac{1}{3} ( x - 4 ) ) + 75}{54 ( x ^{2} - 8 x + 25 )}) + C

Solution steps

\int \frac{8 - 2 x}{( x ^{2} - 8 x + 25 ) ^{2}} d x

Expand

\frac{8 - 2 x}{( x ^{2} - 8 x + 25 ) ^{2}} : \frac{8}{( x ^{2} - 8 x + 25 ) ^{2}} - \frac{2 x}{( x ^{2} - 8 x + 25 ) ^{2}}

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{8}{( x ^{2} - 8 x + 25 ) ^{2}} d x - \int \frac{2 x}{( x ^{2} - 8 x + 25 ) ^{2}} d x

\int \frac{8}{( x ^{2} - 8 x + 25 ) ^{2}} d x = \frac{4}{27} (arctan (\frac{1}{3} (x - 4)) + \frac{1}{2} sin (2 arctan (\frac{1}{3} (x - 4))))

\int \frac{2 x}{( x ^{2} - 8 x + 25 ) ^{2}} d x = 2 (\frac{1}{108} x (2 arctan (\frac{x - 4}{3}) + sin (2 arctan (\frac{x - 4}{3}))) - \frac{x ^{3} arctan ( \frac{1}{3} ( x - 4 ) ) + 3 x ^{2} - 12 x ^{2} arctan ( \frac{1}{3} ( x - 4 ) ) + 57 x arctan ( \frac{1}{3} ( x - 4 ) ) - 24 x - 100 arctan ( \frac{1}{3} ( x - 4 ) ) + 75}{54 ( x ^{2} - 8 x + 25 )})

= \frac{4}{27} (arctan (\frac{1}{3} (x - 4)) + \frac{1}{2} sin (2 arctan (\frac{1}{3} (x - 4)))) - 2 (\frac{1}{108} x (2 arctan (\frac{x - 4}{3}) + sin (2 arctan (\frac{x - 4}{3}))) - \frac{x ^{3} arctan ( \frac{1}{3} ( x - 4 ) ) + 3 x ^{2} - 12 x ^{2} arctan ( \frac{1}{3} ( x - 4 ) ) + 57 x arctan ( \frac{1}{3} ( x - 4 ) ) - 24 x - 100 arctan ( \frac{1}{3} ( x - 4 ) ) + 75}{54 ( x ^{2} - 8 x + 25 )})

Add a constant to the solution

= \frac{4}{27} (arctan (\frac{1}{3} (x - 4)) + \frac{1}{2} sin (2 arctan (\frac{1}{3} (x - 4)))) - 2 (\frac{1}{108} x (2 arctan (\frac{x - 4}{3}) + sin (2 arctan (\frac{x - 4}{3}))) - \frac{x ^{3} arctan ( \frac{1}{3} ( x - 4 ) ) + 3 x ^{2} - 12 x ^{2} arctan ( \frac{1}{3} ( x - 4 ) ) + 57 x arctan ( \frac{1}{3} ( x - 4 ) ) - 24 x - 100 arctan ( \frac{1}{3} ( x - 4 ) ) + 75}{54 ( x ^{2} - 8 x + 25 )}) + C

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What is the integral of (8-2x)/((x^2-8x+25)^2) ?
The integral of (8-2x)/((x^2-8x+25)^2) is