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

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

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

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

Simplify

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

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

Add a constant to the solution

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

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