What is the integral of 1/(\sqrt[3]{tan(x))} ?

The integral of 1/(\sqrt[3]{tan(x))} is

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integral of 1/(\sqrt[3]{tan(x))}

\frac{1}{4} 2 ln tan^{\frac{2}{3}} (x) + 1 - ln \frac{4 tan ^{\frac{4}{3}} ( x ) - 4 tan ^{\frac{2}{3}} ( x ) + 4}{3} + 23 arctan (\frac{2 tan ^{\frac{2}{3}} ( x ) - 1}{3}) + C

Solution steps

Apply u-substitution

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

Take the constant out:

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

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

Factor

\frac{u}{u ^{6} + 1}

= 3 \cdot \int \frac{u}{( u ^{2} + 1 ) ( u ^{4} - u ^{2} + 1 )} d u

Apply u-substitution

= 3 \cdot \int \frac{1}{2 ( v + 1 ) ( v ^{2} - v + 1 )} d v

Take the constant out:

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

= 3 \cdot \frac{1}{2} \cdot \int \frac{1}{( v + 1 ) ( v ^{2} - v + 1 )} d v

Take the partial fraction of

\frac{1}{( v + 1 ) ( v ^{2} - v + 1 )} : \frac{1}{3 ( v + 1 )} + \frac{- v + 2}{3 ( v ^{2} - v + 1 )}

= 3 \cdot \frac{1}{2} \cdot \int \frac{1}{3 ( v + 1 )} + \frac{- v + 2}{3 ( v ^{2} - v + 1 )} d v

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 3 \cdot \frac{1}{2} (\int \frac{1}{3 ( v + 1 )} d v + \int \frac{- v + 2}{3 ( v ^{2} - v + 1 )} d v)

\int \frac{1}{3 ( v + 1 )} d v = \frac{1}{3} ln ∣ v + 1 ∣

\int \frac{- v + 2}{3 ( v ^{2} - v + 1 )} d v = \frac{1}{6} (- ln \frac{4}{3} + \frac{4 v ^{2} - 4 v}{3} + 23 arctan (\frac{2 v - 1}{3}))

= 3 \cdot \frac{1}{2} (\frac{1}{3} ln ∣ v + 1 ∣ + \frac{1}{6} (- ln \frac{4}{3} + \frac{4 v ^{2} - 4 v}{3} + 23 arctan (\frac{2 v - 1}{3})))

Substitute back

Simplify

= \frac{1}{4} 2 ln tan^{\frac{2}{3}} (x) + 1 - ln \frac{4 tan ^{\frac{4}{3}} ( x ) - 4 tan ^{\frac{2}{3}} ( x ) + 4}{3} + 23 arctan (\frac{2 tan ^{\frac{2}{3}} ( x ) - 1}{3})

Add a constant to the solution

= \frac{1}{4} 2 ln tan^{\frac{2}{3}} (x) + 1 - ln \frac{4 tan ^{\frac{4}{3}} ( x ) - 4 tan ^{\frac{2}{3}} ( x ) + 4}{3} + 23 arctan (\frac{2 tan ^{\frac{2}{3}} ( x ) - 1}{3}) + C

What is the integral of 1/(\sqrt[3]{tan(x))} ?
The integral of 1/(\sqrt[3]{tan(x))} is