What is the integral of 1/((x^2+x)(x^2+9)) ?

The integral of 1/((x^2+x)(x^2+9)) is 1/9 ln|x|-1/10 ln|x+1|+1/90 (-1/2 ln|x^2+9|-3arctan(x/3))+C

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integral of 1/((x^2+x)(x^2+9))

\int \frac{1}{( x ^{2} + x ) ( x ^{2} + 9 )} d x

\frac{1}{9} ln ∣ x ∣ - \frac{1}{10} ln ∣ x + 1 ∣ + \frac{1}{90} (- \frac{1}{2} ln x^{2} + 9 - 3 arctan (\frac{x}{3})) + C

Solution steps

\int \frac{1}{( x ^{2} + x ) ( x ^{2} + 9 )} d x

Take the partial fraction of

\frac{1}{( x ^{2} + x ) ( x ^{2} + 9 )} : \frac{1}{9 x} - \frac{1}{10 ( x + 1 )} + \frac{- x - 9}{90 ( x ^{2} + 9 )}

= \int \frac{1}{9 x} - \frac{1}{10 ( x + 1 )} + \frac{- x - 9}{90 ( x ^{2} + 9 )} 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{1}{9 x} d x - \int \frac{1}{10 ( x + 1 )} d x + \int \frac{- x - 9}{90 ( x ^{2} + 9 )} d x

\int \frac{1}{9 x} d x = \frac{1}{9} ln ∣ x ∣

\int \frac{1}{10 ( x + 1 )} d x = \frac{1}{10} ln ∣ x + 1 ∣

\int \frac{- x - 9}{90 ( x ^{2} + 9 )} d x = \frac{1}{90} (- \frac{1}{2} ln x^{2} + 9 - 3 arctan (\frac{x}{3}))

= \frac{1}{9} ln ∣ x ∣ - \frac{1}{10} ln ∣ x + 1 ∣ + \frac{1}{90} (- \frac{1}{2} ln x^{2} + 9 - 3 arctan (\frac{x}{3}))

Add a constant to the solution

= \frac{1}{9} ln ∣ x ∣ - \frac{1}{10} ln ∣ x + 1 ∣ + \frac{1}{90} (- \frac{1}{2} ln x^{2} + 9 - 3 arctan (\frac{x}{3})) + C

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What is the integral of 1/((x^2+x)(x^2+9)) ?
The integral of 1/((x^2+x)(x^2+9)) is 1/9 ln|x|-1/10 ln|x+1|+1/90 (-1/2 ln|x^2+9|-3arctan(x/3))+C