What is the integral of (sec^2(x))/(sqrt(tan(x))) ?

The integral of (sec^2(x))/(sqrt(tan(x))) is

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integral of (sec^2(x))/(sqrt(tan(x)))

\int \frac{sec ^{2} ( x )}{tan ( x )} d x

- 22 (- \frac{1}{8} (ln 2 cot (x) + 22 cot (x) + 2 - 2 arctan (2 cot (x) + 1)) + \frac{1}{8} (ln 2 cot (x) - 22 cot (x) + 2 + 2 arctan (2 cot (x) - 1))) + x tan^{\frac{3}{2}} (x) - arctan (tan (x)) tan^{\frac{3}{2}} (x) - \frac{1}{2 2} ln 2 tan (x) + 22 tan (x) + 2 - \frac{1}{2} arctan (2 tan (x) + 1) + \frac{1}{2 2} ln 2 tan (x) - 22 tan (x) + 2 - \frac{1}{2} arctan (2 tan (x) - 1) + 2 tan (x) + C

Solution steps

\int \frac{sec ^{2} ( x )}{tan ( x )} d x

Rewrite using trig identities

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

Expand

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

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

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

\int \frac{1}{tan ( x )} d x = - 22 (- \frac{1}{8} (ln 2 cot (x) + 22 cot (x) + 2 - 2 arctan (2 cot (x) + 1)) + \frac{1}{8} (ln 2 cot (x) - 22 cot (x) + 2 + 2 arctan (2 cot (x) - 1)))

\int tan^{\frac{3}{2}} (x) d x = x tan^{\frac{3}{2}} (x) - arctan (tan (x)) tan^{\frac{3}{2}} (x) - \frac{1}{2 2} ln 2 tan (x) + 22 tan (x) + 2 - \frac{1}{2} arctan (2 tan (x) + 1) + \frac{1}{2 2} ln 2 tan (x) - 22 tan (x) + 2 - \frac{1}{2} arctan (2 tan (x) - 1) + 2 tan (x)

= - 22 (- \frac{1}{8} (ln 2 cot (x) + 22 cot (x) + 2 - 2 arctan (2 cot (x) + 1)) + \frac{1}{8} (ln 2 cot (x) - 22 cot (x) + 2 + 2 arctan (2 cot (x) - 1))) + x tan^{\frac{3}{2}} (x) - arctan (tan (x)) tan^{\frac{3}{2}} (x) - \frac{1}{2 2} ln 2 tan (x) + 22 tan (x) + 2 - \frac{1}{2} arctan (2 tan (x) + 1) + \frac{1}{2 2} ln 2 tan (x) - 22 tan (x) + 2 - \frac{1}{2} arctan (2 tan (x) - 1) + 2 tan (x)

Add a constant to the solution

= - 22 (- \frac{1}{8} (ln 2 cot (x) + 22 cot (x) + 2 - 2 arctan (2 cot (x) + 1)) + \frac{1}{8} (ln 2 cot (x) - 22 cot (x) + 2 + 2 arctan (2 cot (x) - 1))) + x tan^{\frac{3}{2}} (x) - arctan (tan (x)) tan^{\frac{3}{2}} (x) - \frac{1}{2 2} ln 2 tan (x) + 22 tan (x) + 2 - \frac{1}{2} arctan (2 tan (x) + 1) + \frac{1}{2 2} ln 2 tan (x) - 22 tan (x) + 2 - \frac{1}{2} arctan (2 tan (x) - 1) + 2 tan (x) + C

What is the integral of (sec^2(x))/(sqrt(tan(x))) ?
The integral of (sec^2(x))/(sqrt(tan(x))) is