What is the integral of (sin^2(x))/(1-tan(x)) ?

The integral of (sin^2(x))/(1-tan(x)) is

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integral of (sin^2(x))/(1-tan(x))

\int \frac{sin ^{2} ( x )}{1 - tan ( x )} d x

- \frac{1}{4} ln ∣ cot (x) - 1 ∣ + \frac{1}{8} ln (csc^{2} (x)) + \frac{1}{4} arctan (cot (x)) - \frac{1}{4} (cos (x) sin (x) + arctan (cot (x)) + sin^{2} (x)) + C

Solution steps

\int \frac{sin ^{2} ( x )}{1 - tan ( x )} d x

Apply Trigonometric Substitution

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

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

Take the constant out:

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

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

Take the partial fraction of

\frac{u}{( u - 1 ) ( 1 + u ^{2} ) ^{2}} : \frac{1}{4 ( u - 1 )} + \frac{- u - 1}{4 ( u ^{2} + 1 )} + \frac{- u + 1}{2 ( u ^{2} + 1 ) ^{2}}

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

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

\int \frac{1}{4 ( u - 1 )} d u = \frac{1}{4} ln ∣ u - 1 ∣

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

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

= - (\frac{1}{4} ln ∣ u - 1 ∣ + \frac{1}{4} (- \frac{1}{2} ln u^{2} + 1 - arctan (u)) + \frac{1}{2} (\frac{u}{2 ( u ^{2} + 1 )} + \frac{1}{2} arctan (u) + \frac{1}{2 ( u ^{2} + 1 )}))

Substitute back

u = cot (x)

= - (\frac{1}{4} ln ∣ cot (x) - 1 ∣ + \frac{1}{4} (- \frac{1}{2} ln cot^{2} (x) + 1 - arctan (cot (x))) + \frac{1}{2} (\frac{cot ( x )}{2 ( cot ^{2} ( x ) + 1 )} + \frac{1}{2} arctan (cot (x)) + \frac{1}{2 ( cot ^{2} ( x ) + 1 )}))

Simplify

- (\frac{1}{4} ln ∣ cot (x) - 1 ∣ + \frac{1}{4} (- \frac{1}{2} ln cot^{2} (x) + 1 - arctan (cot (x))) + \frac{1}{2} (\frac{cot ( x )}{2 ( cot ^{2} ( x ) + 1 )} + \frac{1}{2} arctan (cot (x)) + \frac{1}{2 ( cot ^{2} ( x ) + 1 )})) : - \frac{1}{4} ln ∣ cot (x) - 1 ∣ + \frac{1}{8} ln (csc^{2} (x)) + \frac{1}{4} arctan (cot (x)) - \frac{1}{4} (cos (x) sin (x) + arctan (cot (x)) + sin^{2} (x))

= - \frac{1}{4} ln ∣ cot (x) - 1 ∣ + \frac{1}{8} ln (csc^{2} (x)) + \frac{1}{4} arctan (cot (x)) - \frac{1}{4} (cos (x) sin (x) + arctan (cot (x)) + sin^{2} (x))

Add a constant to the solution

= - \frac{1}{4} ln ∣ cot (x) - 1 ∣ + \frac{1}{8} ln (csc^{2} (x)) + \frac{1}{4} arctan (cot (x)) - \frac{1}{4} (cos (x) sin (x) + arctan (cot (x)) + sin^{2} (x)) + C

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What is the integral of (sin^2(x))/(1-tan(x)) ?
The integral of (sin^2(x))/(1-tan(x)) is