What is the integral of x/((x^2+4x+5)^2) ?

The integral of x/((x^2+4x+5)^2) is 1/4 xsin(2arctan(x+2))-arctan(x+2)+C

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integral of x/((x^2+4x+5)^2)

\int \frac{x}{( x ^{2} + 4 x + 5 ) ^{2}} d x

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

Solution steps

\int \frac{x}{( x ^{2} + 4 x + 5 ) ^{2}} d x

Apply Integration By Parts

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

\int \frac{1}{2} (arctan (x + 2) + \frac{1}{2} sin (2 arctan (x + 2))) d x = \frac{1}{2} arctan (2 + x) (2 + x)

= \frac{1}{2} x (arctan (x + 2) + \frac{1}{2} sin (2 arctan (x + 2))) - \frac{1}{2} arctan (2 + x) (2 + x)

Simplify

\frac{1}{2} x (arctan (x + 2) + \frac{1}{2} sin (2 arctan (x + 2))) - \frac{1}{2} arctan (2 + x) (2 + x) : \frac{1}{4} x sin (2 arctan (x + 2)) - arctan (x + 2)

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

Add a constant to the solution

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

What is the integral of x/((x^2+4x+5)^2) ?
The integral of x/((x^2+4x+5)^2) is 1/4 xsin(2arctan(x+2))-arctan(x+2)+C