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

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

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integral of (x+x^3)/(1+x^4)

\int \frac{x + x ^{3}}{1 + x ^{4}} d x

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

Solution steps

\int \frac{x + x ^{3}}{1 + x ^{4}} d x

Expand

\frac{x + x ^{3}}{1 + x ^{4}} : \frac{x}{1 + x ^{4}} + \frac{x ^{3}}{1 + x ^{4}}

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{x}{1 + x ^{4}} d x + \int \frac{x ^{3}}{1 + x ^{4}} d x

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

\int \frac{x ^{3}}{1 + x ^{4}} d x = \frac{1}{4} ln 1 + x^{4}

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

Add a constant to the solution

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

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