What is the integral of (x+5)/((x^2-6x+13)^2) ?

The integral of (x+5)/((x^2-6x+13)^2) is

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integral of (x+5)/((x^2-6x+13)^2)

\int \frac{x + 5}{( x ^{2} - 6 x + 13 ) ^{2}} d x

\frac{1}{16} x arctan (\frac{1}{2} (x - 3)) + \frac{1}{32} x sin (2 arctan (\frac{1}{2} (x - 3))) - \frac{1}{16} x arctan (\frac{1}{2} (x - 3)) + \frac{1}{2} arctan (\frac{1}{2} (x - 3)) + \frac{5}{32} sin (2 arctan (\frac{1}{2} (x - 3))) + C

Solution steps

\int \frac{x + 5}{( x ^{2} - 6 x + 13 ) ^{2}} d x

Expand

\frac{x + 5}{( x ^{2} - 6 x + 13 ) ^{2}} : \frac{x}{( x ^{2} - 6 x + 13 ) ^{2}} + \frac{5}{( x ^{2} - 6 x + 13 ) ^{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{x}{( x ^{2} - 6 x + 13 ) ^{2}} d x + \int \frac{5}{( x ^{2} - 6 x + 13 ) ^{2}} d x

\int \frac{x}{( x ^{2} - 6 x + 13 ) ^{2}} d x = \frac{1}{32} x (2 arctan (\frac{x - 3}{2}) + sin (2 arctan (\frac{x - 3}{2}))) - \frac{1}{16} arctan (\frac{1}{2} (x - 3)) (x - 3)

\int \frac{5}{( x ^{2} - 6 x + 13 ) ^{2}} d x = \frac{5}{16} (arctan (\frac{1}{2} (x - 3)) + \frac{1}{2} sin (2 arctan (\frac{1}{2} (x - 3))))

= \frac{1}{32} x (2 arctan (\frac{x - 3}{2}) + sin (2 arctan (\frac{x - 3}{2}))) - \frac{1}{16} arctan (\frac{1}{2} (x - 3)) (x - 3) + \frac{5}{16} (arctan (\frac{1}{2} (x - 3)) + \frac{1}{2} sin (2 arctan (\frac{1}{2} (x - 3))))

Simplify

\frac{1}{32} x (2 arctan (\frac{x - 3}{2}) + sin (2 arctan (\frac{x - 3}{2}))) - \frac{1}{16} arctan (\frac{1}{2} (x - 3)) (x - 3) + \frac{5}{16} (arctan (\frac{1}{2} (x - 3)) + \frac{1}{2} sin (2 arctan (\frac{1}{2} (x - 3)))) : \frac{1}{16} x arctan (\frac{1}{2} (x - 3)) + \frac{1}{32} x sin (2 arctan (\frac{1}{2} (x - 3))) - \frac{1}{16} x arctan (\frac{1}{2} (x - 3)) + \frac{1}{2} arctan (\frac{1}{2} (x - 3)) + \frac{5}{32} sin (2 arctan (\frac{1}{2} (x - 3)))

= \frac{1}{16} x arctan (\frac{1}{2} (x - 3)) + \frac{1}{32} x sin (2 arctan (\frac{1}{2} (x - 3))) - \frac{1}{16} x arctan (\frac{1}{2} (x - 3)) + \frac{1}{2} arctan (\frac{1}{2} (x - 3)) + \frac{5}{32} sin (2 arctan (\frac{1}{2} (x - 3)))

Add a constant to the solution

= \frac{1}{16} x arctan (\frac{1}{2} (x - 3)) + \frac{1}{32} x sin (2 arctan (\frac{1}{2} (x - 3))) - \frac{1}{16} x arctan (\frac{1}{2} (x - 3)) + \frac{1}{2} arctan (\frac{1}{2} (x - 3)) + \frac{5}{32} sin (2 arctan (\frac{1}{2} (x - 3))) + C

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What is the integral of (x+5)/((x^2-6x+13)^2) ?
The integral of (x+5)/((x^2-6x+13)^2) is