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

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

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

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

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

Solution steps

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

Take the partial fraction of

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

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

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

\int \frac{1}{2 ( x + 1 )} d x = \frac{1}{2} ln ∣ x + 1 ∣

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

Add a constant to the solution

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

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