What is the integral of (x^3+x^2+5x+4)/((x^2+4)^2) ?

The integral of (x^3+x^2+5x+4)/((x^2+4)^2) is

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integral of (x^3+x^2+5x+4)/((x^2+4)^2)

\int \frac{x ^{3} + x ^{2} + 5 x + 4}{( x ^{2} + 4 ) ^{2}} d x

- \frac{x}{2 ( x ^{2} + 4 )} + \frac{1}{2} ln x^{2} + 4 + \frac{1}{2} arctan (\frac{x}{2}) + \frac{1}{8} sin (2 arctan (\frac{x}{2})) + \frac{2}{x ^{2} + 4} - \frac{5}{2 ( x ^{2} + 4 )} + C

Solution steps

\int \frac{x ^{3} + x ^{2} + 5 x + 4}{( x ^{2} + 4 ) ^{2}} d x

Expand

\frac{x ^{3} + x ^{2} + 5 x + 4}{( x ^{2} + 4 ) ^{2}} : \frac{x ^{3}}{( x ^{2} + 4 ) ^{2}} + \frac{x ^{2}}{( x ^{2} + 4 ) ^{2}} + \frac{5 x}{( x ^{2} + 4 ) ^{2}} + \frac{4}{( x ^{2} + 4 ) ^{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 ^{3}}{( x ^{2} + 4 ) ^{2}} d x + \int \frac{x ^{2}}{( x ^{2} + 4 ) ^{2}} d x + \int \frac{5 x}{( x ^{2} + 4 ) ^{2}} d x + \int \frac{4}{( x ^{2} + 4 ) ^{2}} d x

\int \frac{x ^{3}}{( x ^{2} + 4 ) ^{2}} d x = \frac{1}{2} (ln x^{2} + 4 + \frac{4}{x ^{2} + 4})

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

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

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

= \frac{1}{2} (ln x^{2} + 4 + \frac{4}{x ^{2} + 4}) + \frac{1}{2} (\frac{1}{2} arctan (\frac{x}{2}) - \frac{x}{4 + x ^{2}}) - \frac{5}{2 ( x ^{2} + 4 )} + \frac{1}{4} (arctan (\frac{x}{2}) + \frac{1}{2} sin (2 arctan (\frac{x}{2})))

Simplify

\frac{1}{2} (ln x^{2} + 4 + \frac{4}{x ^{2} + 4}) + \frac{1}{2} (\frac{1}{2} arctan (\frac{x}{2}) - \frac{x}{4 + x ^{2}}) - \frac{5}{2 ( x ^{2} + 4 )} + \frac{1}{4} (arctan (\frac{x}{2}) + \frac{1}{2} sin (2 arctan (\frac{x}{2}))) : - \frac{x}{2 ( x ^{2} + 4 )} + \frac{1}{2} ln x^{2} + 4 + \frac{1}{2} arctan (\frac{x}{2}) + \frac{1}{8} sin (2 arctan (\frac{x}{2})) + \frac{2}{x ^{2} + 4} - \frac{5}{2 ( x ^{2} + 4 )}

= - \frac{x}{2 ( x ^{2} + 4 )} + \frac{1}{2} ln x^{2} + 4 + \frac{1}{2} arctan (\frac{x}{2}) + \frac{1}{8} sin (2 arctan (\frac{x}{2})) + \frac{2}{x ^{2} + 4} - \frac{5}{2 ( x ^{2} + 4 )}

Add a constant to the solution

= - \frac{x}{2 ( x ^{2} + 4 )} + \frac{1}{2} ln x^{2} + 4 + \frac{1}{2} arctan (\frac{x}{2}) + \frac{1}{8} sin (2 arctan (\frac{x}{2})) + \frac{2}{x ^{2} + 4} - \frac{5}{2 ( x ^{2} + 4 )} + C

What is the integral of (x^3+x^2+5x+4)/((x^2+4)^2) ?
The integral of (x^3+x^2+5x+4)/((x^2+4)^2) is