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

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

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

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

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

Solution steps

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

Expand

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

\int \frac{4 x}{1 + x ^{2}} d x = 2 ln 1 + x^{2}

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

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

Add a constant to the solution

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

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