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

The integral of (arctan^3(x))/(3x^2+3) is 1/3 arctan^4(x)-(arctan^4(x))/4+C

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integral of (arctan^3(x))/(3x^2+3)

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

= \frac{1}{3} arctan^{4} (x) - \frac{arctan ^{4} ( x )}{4}

Add a constant to the solution

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

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What is the integral of (arctan^3(x))/(3x^2+3) ?
The integral of (arctan^3(x))/(3x^2+3) is 1/3 arctan^4(x)-(arctan^4(x))/4+C