What is the integral of (cos^3(x))/(sin(x)+cos(x)) ?

The integral of (cos^3(x))/(sin(x)+cos(x)) is

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integral of (cos^3(x))/(sin(x)+cos(x))

\int \frac{cos ^{3} ( x )}{sin ( x ) + cos ( x )} d x

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

Solution steps

\int \frac{cos ^{3} ( x )}{sin ( x ) + cos ( x )} d x

Apply u-substitution

: \int \frac{2 ( 1 - u ^{2} ) ^{3}}{( - u ^{2} + 2 u + 1 ) ( u ^{2} + 1 ) ^{3}} d u

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

Take the constant out:

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

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

Take the partial fraction of

\frac{( 1 - u ^{2} ) ^{3}}{( - u ^{2} + 2 u + 1 ) ( u ^{2} + 1 ) ^{3}} : \frac{u - 1}{4 ( u ^{2} - 2 u - 1 )} + \frac{- u + 3}{4 ( u ^{2} + 1 )} + \frac{u - 2}{( u ^{2} + 1 ) ^{2}} + \frac{- 2 u + 2}{( u ^{2} + 1 ) ^{3}}

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

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

\int \frac{u - 1}{4 ( u ^{2} - 2 u - 1 )} d u = \frac{1}{8} ln u^{2} - 2 u - 1

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

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

\int \frac{- 2 u + 2}{( u ^{2} + 1 ) ^{3}} d u = \frac{1}{2 ( u ^{2} + 1 ) ^{2}} + 2 (\frac{3}{8} (arctan (u) + \frac{u}{1 + u ^{2}}) + \frac{u}{4 ( u ^{2} + 1 ) ^{2}})

= 2 (\frac{1}{8} ln u^{2} - 2 u - 1 + \frac{1}{4} (- \frac{1}{2} ln u^{2} + 1 + 3 arctan (u)) - arctan (u) - \frac{u}{u ^{2} + 1} - \frac{1}{2 ( u ^{2} + 1 )} + \frac{1}{2 ( u ^{2} + 1 ) ^{2}} + 2 (\frac{3}{8} (arctan (u) + \frac{u}{1 + u ^{2}}) + \frac{u}{4 ( u ^{2} + 1 ) ^{2}}))

Substitute back

u = tan (\frac{x}{2})

= 2 (\frac{1}{8} ln tan^{2} (\frac{x}{2}) - 2 tan (\frac{x}{2}) - 1 + \frac{1}{4} (- \frac{1}{2} ln tan^{2} (\frac{x}{2}) + 1 + 3 arctan (tan (\frac{x}{2}))) - arctan (tan (\frac{x}{2})) - \frac{tan ( \frac{x}{2} )}{tan ^{2} ( \frac{x}{2} ) + 1} - \frac{1}{2 ( tan ^{2} ( \frac{x}{2} ) + 1 )} + \frac{1}{2 ( tan ^{2} ( \frac{x}{2} ) + 1 ) ^{2}} + 2 (\frac{3}{8} (arctan (tan (\frac{x}{2})) + \frac{tan ( \frac{x}{2} )}{1 + tan ^{2} ( \frac{x}{2} )}) + \frac{tan ( \frac{x}{2} )}{4 ( tan ^{2} ( \frac{x}{2} ) + 1 ) ^{2}}))

Simplify

2 (\frac{1}{8} ln tan^{2} (\frac{x}{2}) - 2 tan (\frac{x}{2}) - 1 + \frac{1}{4} (- \frac{1}{2} ln tan^{2} (\frac{x}{2}) + 1 + 3 arctan (tan (\frac{x}{2}))) - arctan (tan (\frac{x}{2})) - \frac{tan ( \frac{x}{2} )}{tan ^{2} ( \frac{x}{2} ) + 1} - \frac{1}{2 ( tan ^{2} ( \frac{x}{2} ) + 1 )} + \frac{1}{2 ( tan ^{2} ( \frac{x}{2} ) + 1 ) ^{2}} + 2 (\frac{3}{8} (arctan (tan (\frac{x}{2})) + \frac{tan ( \frac{x}{2} )}{1 + tan ^{2} ( \frac{x}{2} )}) + \frac{tan ( \frac{x}{2} )}{4 ( tan ^{2} ( \frac{x}{2} ) + 1 ) ^{2}})) : \frac{1}{4} ln tan^{2} (\frac{x}{2}) - 2 tan (\frac{x}{2}) - 1 - \frac{1}{4} ln (sec^{2} (\frac{x}{2})) + arctan (tan (\frac{x}{2})) - \frac{1}{2} cos^{2} (\frac{x}{2}) tan (\frac{x}{2}) - cos^{2} (\frac{x}{2}) + cos^{4} (\frac{x}{2}) + cos^{4} (\frac{x}{2}) tan (\frac{x}{2})

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

Add a constant to the solution

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

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What is the integral of (cos^3(x))/(sin(x)+cos(x)) ?
The integral of (cos^3(x))/(sin(x)+cos(x)) is