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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

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

= 3 (- \frac{1}{2} ln ∣ x + 1 ∣ - \frac{1}{2 ( x + 1 )} + \frac{1}{4} ln x^{2} + 1) + \frac{1}{x + 1} + arctan (x) + \frac{1}{2} ln ∣ x + 1 ∣ - \frac{1}{2 ( x + 1 )} - \frac{1}{4} ln x^{2} + 1

Simplify

3 (- \frac{1}{2} ln ∣ x + 1 ∣ - \frac{1}{2 ( x + 1 )} + \frac{1}{4} ln x^{2} + 1) + \frac{1}{x + 1} + arctan (x) + \frac{1}{2} ln ∣ x + 1 ∣ - \frac{1}{2 ( x + 1 )} - \frac{1}{4} ln x^{2} + 1 : - ln ∣ x + 1 ∣ + \frac{1}{2} ln x^{2} + 1 - \frac{1}{x + 1} + arctan (x)

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

Add a constant to the solution

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

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