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

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

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

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

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

Solution steps

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

Expand

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

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

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

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

Add a constant to the solution

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

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