What is the integral of ((x/3+1/3))/((x^2+x+1)^2) ?

The integral of ((x/3+1/3))/((x^2+x+1)^2) is

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integral of ((x/3+1/3))/((x^2+x+1)^2)

\int \frac{( \frac{x}{3} + \frac{1}{3} )}{( x ^{2} + x + 1 ) ^{2}} d x

\frac{1}{3} (8 (- \frac{1}{4 ( 4 x ^{2} + 4 x + 4 )} - \frac{1}{24 3} (2 arctan (\frac{1}{3} (2 x + 1)) + sin (2 arctan (\frac{1}{3} (2 x + 1))))) + \frac{2}{3 3} (2 arctan (\frac{2 x + 1}{3}) + sin (2 arctan (\frac{2 x + 1}{3})))) + C

Solution steps

\int \frac{\frac{x}{3} + \frac{1}{3}}{( x ^{2} + x + 1 ) ^{2}} d x

Simplify

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

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{3} \cdot \int \frac{x + 1}{( x ^{2} + x + 1 ) ^{2}} d x

Expand

\frac{x + 1}{( x ^{2} + x + 1 ) ^{2}} : \frac{x}{( x ^{2} + x + 1 ) ^{2}} + \frac{1}{( x ^{2} + x + 1 ) ^{2}}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int \frac{x}{( x ^{2} + x + 1 ) ^{2}} d x = 8 (- \frac{1}{4 ( 4 x ^{2} + 4 x + 4 )} - \frac{1}{24 3} (2 arctan (\frac{1}{3} (2 x + 1)) + sin (2 arctan (\frac{1}{3} (2 x + 1)))))

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

= \frac{1}{3} (8 (- \frac{1}{4 ( 4 x ^{2} + 4 x + 4 )} - \frac{1}{24 3} (2 arctan (\frac{1}{3} (2 x + 1)) + sin (2 arctan (\frac{1}{3} (2 x + 1))))) + \frac{2}{3 3} (2 arctan (\frac{2 x + 1}{3}) + sin (2 arctan (\frac{2 x + 1}{3}))))

Add a constant to the solution

= \frac{1}{3} (8 (- \frac{1}{4 ( 4 x ^{2} + 4 x + 4 )} - \frac{1}{24 3} (2 arctan (\frac{1}{3} (2 x + 1)) + sin (2 arctan (\frac{1}{3} (2 x + 1))))) + \frac{2}{3 3} (2 arctan (\frac{2 x + 1}{3}) + sin (2 arctan (\frac{2 x + 1}{3})))) + C

\int x^{3} 1 + (3 x^{2})^{2} d x

\int \frac{x ln ( x ^{2} + 3 )}{x ^{2} + 3} d x

\int 1^{2} - x^{2} d x

\int \frac{1}{x ^{2} x + 1} d x

\int \frac{1}{x + 9} d x

What is the integral of ((x/3+1/3))/((x^2+x+1)^2) ?
The integral of ((x/3+1/3))/((x^2+x+1)^2) is