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

The integral of 1/x (arctan(1/x)) is -1/2 i(\Li_{2}(-i 1/x)-\Li_{2}(i 1/x))+C

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

integral

\frac{1}{x} (arctan (\frac{1}{x}))

- \frac{1}{2} i (Li_{2} (- i \frac{1}{x}) - Li_{2} (i \frac{1}{x})) + C

Solution steps

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

Apply u-substitution

: \int - \frac{arctan ( u )}{u} d u

= \int - \frac{arctan ( u )}{u} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int \frac{arctan ( u )}{u} d u

Use the nonelementary integral:

\int \frac{arctan ( x )}{x} d x = \frac{1}{2} i (Li_{2} (- i x) - Li_{2} (i x))

= - \frac{1}{2} i (Li_{2} (- i u) - Li_{2} (i u))

Substitute back

u = \frac{1}{x}

= - \frac{1}{2} i (Li_{2} (- i \frac{1}{x}) - Li_{2} (i \frac{1}{x}))

Add a constant to the solution

= - \frac{1}{2} i (Li_{2} (- i \frac{1}{x}) - Li_{2} (i \frac{1}{x})) + C

- 4 e^{3} x

tan (\frac{x}{2})

(2 x - x^{2} + 7) e^{2} x

ln (x^{2} + 1) + 6

\frac{1}{( t ^{2} + t )}

What is the integral of 1/x (arctan(1/x)) ?
The integral of 1/x (arctan(1/x)) is -1/2 i(\Li_{2}(-i 1/x)-\Li_{2}(i 1/x))+C