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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

Add a constant to the solution

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

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