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

The integral of xarctan(x)ln(1+x^2) is

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

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

Simplify

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

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

Add a constant to the solution

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

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