What is the integral of (ln(x))/((1+x^2) 3/2) ?

The integral of (ln(x))/((1+x^2) 3/2) is 1/3 (2ln(x)arctan(x)-i(\Li_{2}(-ix)-\Li_{2}(ix)))+C

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integral of (ln(x))/((1+x^2) 3/2)

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

\frac{1}{3} (2 ln (x) arctan (x) - i (Li_{2} (- i x) - Li_{2} (i x))) + C

Solution steps

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

Take the constant out:

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

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

Simplify

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

Apply Integration By Parts

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

= \frac{2}{3} (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))

= \frac{2}{3} (ln (x) arctan (x) - \frac{1}{2} i (Li_{2} (- i x) - Li_{2} (i x)))

Simplify

\frac{2}{3} (ln (x) arctan (x) - \frac{1}{2} i (Li_{2} (- i x) - Li_{2} (i x))) : \frac{1}{3} (2 ln (x) arctan (x) - i (Li_{2} (- i x) - Li_{2} (i x)))

= \frac{1}{3} (2 ln (x) arctan (x) - i (Li_{2} (- i x) - Li_{2} (i x)))

Add a constant to the solution

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

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What is the integral of (ln(x))/((1+x^2) 3/2) ?
The integral of (ln(x))/((1+x^2) 3/2) is 1/3 (2ln(x)arctan(x)-i(\Li_{2}(-ix)-\Li_{2}(ix)))+C