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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

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

Add a constant to the solution

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

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