What is the integral of (arctan(5x))/x ?

The integral of (arctan(5x))/x is 1/2 i(\Li_{2}(-i5x)-\Li_{2}(i5x))+C

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integral of (arctan(5x))/x

\int \frac{arctan ( 5 x )}{x} d x

\frac{1}{2} i (Li_{2} (- i 5 x) - Li_{2} (i 5 x)) + C

Solution steps

\int \frac{arctan ( 5 x )}{x} d x

Apply u-substitution

= \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 = 5 x

= \frac{1}{2} i (Li_{2} (- i 5 x) - Li_{2} (i 5 x))

Add a constant to the solution

= \frac{1}{2} i (Li_{2} (- i 5 x) - Li_{2} (i 5 x)) + C

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What is the integral of (arctan(5x))/x ?
The integral of (arctan(5x))/x is 1/2 i(\Li_{2}(-i5x)-\Li_{2}(i5x))+C