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

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

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

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

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

Solution steps

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

Apply Integration By Parts

: ln (x) arctan (x) - \int \frac{arctan ( x )}{x} d x

= ln (x) arctan (x) - \int \frac{arctan ( x )}{x} d x

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

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

Add a constant to the solution

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

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