What is the integral of (arctan(1/x))/(x^4) ?

The integral of (arctan(1/x))/(x^4) is -(arctan(1/x))/(3x^3)-1/3 (-ln|x|-1/(2x^2)+1/2 ln|x^2+1|)+C

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integral of (arctan(1/x))/(x^4)

\int \frac{arctan ( \frac{1}{x} )}{x ^{4}} d x

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

Solution steps

\int \frac{arctan ( \frac{1}{x} )}{x ^{4}} d x

Apply Integration By Parts

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

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

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

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

Add a constant to the solution

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

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What is the integral of (arctan(1/x))/(x^4) ?
The integral of (arctan(1/x))/(x^4) is -(arctan(1/x))/(3x^3)-1/3 (-ln|x|-1/(2x^2)+1/2 ln|x^2+1|)+C