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

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

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

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

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

Solution steps

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

Apply u-substitution

: \int \frac{1}{6 u ^{\frac{5}{3}}} d u

= \int \frac{1}{6 u ^{\frac{5}{3}}} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{6} \cdot \int \frac{1}{u ^{\frac{5}{3}}} d u

Apply the Power Rule

: - \frac{3}{2 u ^{\frac{2}{3}}}

= \frac{1}{6} (- \frac{3}{2 u ^{\frac{2}{3}}})

Substitute back

u = arctan^{3} (x^{2})

= \frac{1}{6} - \frac{3}{2 ( arctan ^{3} ( x ^{2} ) ) ^{\frac{2}{3}}}

Simplify

\frac{1}{6} - \frac{3}{2 ( arctan ^{3} ( x ^{2} ) ) ^{\frac{2}{3}}} : - \frac{1}{4 arctan ^{2} ( x ^{2} )}

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

Add a constant to the solution

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

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