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

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

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

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

arctan (arctan (x)) + C

Solution steps

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

Apply u-substitution

: \int \frac{1}{1 + u ^{2}} d u

= \int \frac{1}{1 + u ^{2}} d u

Use the common integral:

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

= arctan (u)

Substitute back

u = arctan (x)

= arctan (arctan (x))

Add a constant to the solution

= arctan (arctan (x)) + C

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