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

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

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

integral

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

\frac{x ^{4} arctan ( \frac{x}{3} ) + 3 x ^{3} + 6 x ^{2} arctan ( \frac{x}{3} ) + 5 3 x + 9 arctan ( \frac{x}{3} )}{24 3 ( x ^{2} + 3 ) ^{2}} + C

Solution steps

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

Apply exponent rule:

\frac{1}{a ^{b}} = a^{- b}

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

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

Apply Integral Substitution

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

Take the constant out:

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

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

Factor

\frac{1}{( 3 u ^{2} + 3 ) ^{3}}

= 3 \cdot \int \frac{1}{27 ( u ^{2} + 1 ) ^{3}} d u

Take the constant out:

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

= 3 \frac{1}{27} \cdot \int \frac{1}{( u ^{2} + 1 ) ^{3}} d u

Apply Integral Reduction

= 3 \frac{1}{27} (\frac{3}{4} \cdot \int \frac{1}{( u ^{2} + 1 ) ^{2}} d u + \frac{u}{2 \cdot 2 ( u ^{2} + 1 ) ^{2}})

\int \frac{1}{( u ^{2} + 1 ) ^{2}} d u = \frac{1}{2} (arctan (u) + \frac{u}{1 + u ^{2}})

= 3 \frac{1}{27} (\frac{3}{4} \cdot \frac{1}{2} (arctan (u) + \frac{u}{1 + u ^{2}}) + \frac{u}{2 \cdot 2 ( u ^{2} + 1 ) ^{2}})

Substitute back

u = \frac{x}{3}

= 3 \frac{1}{27} \frac{3}{4} \cdot \frac{1}{2} arctan (\frac{x}{3}) + \frac{\frac{x}{3}}{1 + ( \frac{x}{3} ) ^{2}} + \frac{\frac{x}{3}}{2 \cdot 2 ( ( \frac{x}{3} ) ^{2} + 1 ) ^{2}}

Simplify

3 \frac{1}{27} \frac{3}{4} \cdot \frac{1}{2} arctan (\frac{x}{3}) + \frac{\frac{x}{3}}{1 + ( \frac{x}{3} ) ^{2}} + \frac{\frac{x}{3}}{2 \cdot 2 ( ( \frac{x}{3} ) ^{2} + 1 ) ^{2}} : \frac{x ^{4} arctan ( \frac{x}{3} ) + 3 x ^{3} + 6 x ^{2} arctan ( \frac{x}{3} ) + 5 3 x + 9 arctan ( \frac{x}{3} )}{24 3 ( x ^{2} + 3 ) ^{2}}

= \frac{x ^{4} arctan ( \frac{x}{3} ) + 3 x ^{3} + 6 x ^{2} arctan ( \frac{x}{3} ) + 5 3 x + 9 arctan ( \frac{x}{3} )}{24 3 ( x ^{2} + 3 ) ^{2}}

Add a constant to the solution

= \frac{x ^{4} arctan ( \frac{x}{3} ) + 3 x ^{3} + 6 x ^{2} arctan ( \frac{x}{3} ) + 5 3 x + 9 arctan ( \frac{x}{3} )}{24 3 ( x ^{2} + 3 ) ^{2}} + C