What is the integral of (x^4+x^3+1)/((x^2+4)^3) ?

The integral of (x^4+x^3+1)/((x^2+4)^3) is

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integral of (x^4+x^3+1)/((x^2+4)^3)

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

- \frac{1}{4} sin (2 arctan (\frac{x}{2})) + \frac{3 x ^{4} arctan ( \frac{x}{2} ) + 6 x ^{3} + 24 x ^{2} arctan ( \frac{x}{2} ) + 40 x + 48 arctan ( \frac{x}{2} )}{16 ( x ^{2} + 4 ) ^{2}} - \frac{1}{2 ( x ^{2} + 4 )} + \frac{1}{( x ^{2} + 4 ) ^{2}} + \frac{3 x ^{4} arctan ( \frac{x}{2} ) + 6 x ^{3} + 24 x ^{2} arctan ( \frac{x}{2} ) + 40 x + 48 arctan ( \frac{x}{2} )}{256 ( x ^{2} + 4 ) ^{2}} + C

Solution steps

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

Expand

\frac{x ^{4} + x ^{3} + 1}{( x ^{2} + 4 ) ^{3}} : \frac{x ^{4}}{( x ^{2} + 4 ) ^{3}} + \frac{x ^{3}}{( x ^{2} + 4 ) ^{3}} + \frac{1}{( x ^{2} + 4 ) ^{3}}

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

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

\int \frac{x ^{4}}{( x ^{2} + 4 ) ^{3}} d x = - \frac{1}{4} sin (2 arctan (\frac{x}{2})) + \frac{3 x ^{4} arctan ( \frac{x}{2} ) + 6 x ^{3} + 24 x ^{2} arctan ( \frac{x}{2} ) + 40 x + 48 arctan ( \frac{x}{2} )}{16 ( x ^{2} + 4 ) ^{2}}

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

\int \frac{1}{( x ^{2} + 4 ) ^{3}} d x = \frac{3 x ^{4} arctan ( \frac{x}{2} ) + 6 x ^{3} + 24 x ^{2} arctan ( \frac{x}{2} ) + 40 x + 48 arctan ( \frac{x}{2} )}{256 ( x ^{2} + 4 ) ^{2}}

= - \frac{1}{4} sin (2 arctan (\frac{x}{2})) + \frac{3 x ^{4} arctan ( \frac{x}{2} ) + 6 x ^{3} + 24 x ^{2} arctan ( \frac{x}{2} ) + 40 x + 48 arctan ( \frac{x}{2} )}{16 ( x ^{2} + 4 ) ^{2}} - \frac{1}{2 ( x ^{2} + 4 )} + \frac{1}{( x ^{2} + 4 ) ^{2}} + \frac{3 x ^{4} arctan ( \frac{x}{2} ) + 6 x ^{3} + 24 x ^{2} arctan ( \frac{x}{2} ) + 40 x + 48 arctan ( \frac{x}{2} )}{256 ( x ^{2} + 4 ) ^{2}}

Add a constant to the solution

= - \frac{1}{4} sin (2 arctan (\frac{x}{2})) + \frac{3 x ^{4} arctan ( \frac{x}{2} ) + 6 x ^{3} + 24 x ^{2} arctan ( \frac{x}{2} ) + 40 x + 48 arctan ( \frac{x}{2} )}{16 ( x ^{2} + 4 ) ^{2}} - \frac{1}{2 ( x ^{2} + 4 )} + \frac{1}{( x ^{2} + 4 ) ^{2}} + \frac{3 x ^{4} arctan ( \frac{x}{2} ) + 6 x ^{3} + 24 x ^{2} arctan ( \frac{x}{2} ) + 40 x + 48 arctan ( \frac{x}{2} )}{256 ( x ^{2} + 4 ) ^{2}} + C

What is the integral of (x^4+x^3+1)/((x^2+4)^3) ?
The integral of (x^4+x^3+1)/((x^2+4)^3) is