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

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

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

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

- \frac{ln ( x ^{2} + x + 1 )}{2 x ^{2}} - \frac{1}{4} ln 4 x^{2} + 4 x + 4 - \frac{3}{2} arctan (\frac{1}{3} (2 x + 1)) + \frac{1}{2} ln ∣ x ∣ - \frac{1}{2 x} + C

Solution steps

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

Apply Integration By Parts

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

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

\int - \frac{2 x + 1}{2 x ^{2} ( x ^{2} + x + 1 )} d x = \frac{1}{4} ln 4 x^{2} + 4 x + 4 + \frac{1}{3} arctan (\frac{1}{3} (2 x + 1)) + \frac{1}{2 3} arctan (\frac{1}{3} (2 x + 1)) - \frac{1}{2} ln ∣ x ∣ + \frac{1}{2 x}

= - \frac{ln ( x ^{2} + x + 1 )}{2 x ^{2}} - (\frac{1}{4} ln 4 x^{2} + 4 x + 4 + \frac{1}{3} arctan (\frac{1}{3} (2 x + 1)) + \frac{1}{2 3} arctan (\frac{1}{3} (2 x + 1)) - \frac{1}{2} ln ∣ x ∣ + \frac{1}{2 x})

Simplify

- \frac{ln ( x ^{2} + x + 1 )}{2 x ^{2}} - (\frac{1}{4} ln 4 x^{2} + 4 x + 4 + \frac{1}{3} arctan (\frac{1}{3} (2 x + 1)) + \frac{1}{2 3} arctan (\frac{1}{3} (2 x + 1)) - \frac{1}{2} ln ∣ x ∣ + \frac{1}{2 x}) : - \frac{ln ( x ^{2} + x + 1 )}{2 x ^{2}} - \frac{1}{4} ln 4 x^{2} + 4 x + 4 - \frac{3}{2} arctan (\frac{1}{3} (2 x + 1)) + \frac{1}{2} ln ∣ x ∣ - \frac{1}{2 x}

= - \frac{ln ( x ^{2} + x + 1 )}{2 x ^{2}} - \frac{1}{4} ln 4 x^{2} + 4 x + 4 - \frac{3}{2} arctan (\frac{1}{3} (2 x + 1)) + \frac{1}{2} ln ∣ x ∣ - \frac{1}{2 x}

Add a constant to the solution

= - \frac{ln ( x ^{2} + x + 1 )}{2 x ^{2}} - \frac{1}{4} ln 4 x^{2} + 4 x + 4 - \frac{3}{2} arctan (\frac{1}{3} (2 x + 1)) + \frac{1}{2} ln ∣ x ∣ - \frac{1}{2 x} + C

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