What is the integral of (cos^3(x))/(sin^3(x)+8) ?

The integral of (cos^3(x))/(sin^3(x)+8) is

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integral of (cos^3(x))/(sin^3(x)+8)

\int \frac{cos ^{3} ( x )}{sin ^{3} ( x ) + 8} d x

\frac{1}{12} ln ∣ sin (x) + 2 ∣ + \frac{1}{24} (- ln \frac{1}{3} (sin (x) - 1)^{2} + 1 + 23 arctan (\frac{1}{3} (sin (x) - 1))) - \frac{1}{3} ln sin^{3} (x) + 8 + C

Solution steps

\int \frac{cos ^{3} ( x )}{sin ^{3} ( x ) + 8} d x

Apply u-substitution

: \int \frac{1 - u ^{2}}{u ^{3} + 8} d u

= \int \frac{1 - u ^{2}}{u ^{3} + 8} d u

Expand

\frac{1 - u ^{2}}{u ^{3} + 8} : \frac{1}{u ^{3} + 8} - \frac{u ^{2}}{u ^{3} + 8}

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}{u ^{3} + 8} d u - \int \frac{u ^{2}}{u ^{3} + 8} d u

\int \frac{1}{u ^{3} + 8} d u = \frac{1}{12} ln ∣ u + 2 ∣ + \frac{1}{12} (- \frac{1}{2} ln \frac{1}{3} (u - 1)^{2} + 1 + 3 arctan (\frac{1}{3} (u - 1)))

\int \frac{u ^{2}}{u ^{3} + 8} d u = \frac{1}{3} ln u^{3} + 8

= \frac{1}{12} ln ∣ u + 2 ∣ + \frac{1}{12} (- \frac{1}{2} ln \frac{1}{3} (u - 1)^{2} + 1 + 3 arctan (\frac{1}{3} (u - 1))) - \frac{1}{3} ln u^{3} + 8

Substitute back

u = sin (x)

= \frac{1}{12} ln ∣ sin (x) + 2 ∣ + \frac{1}{12} (- \frac{1}{2} ln \frac{1}{3} (sin (x) - 1)^{2} + 1 + 3 arctan (\frac{1}{3} (sin (x) - 1))) - \frac{1}{3} ln sin^{3} (x) + 8

Simplify

\frac{1}{12} ln ∣ sin (x) + 2 ∣ + \frac{1}{12} (- \frac{1}{2} ln \frac{1}{3} (sin (x) - 1)^{2} + 1 + 3 arctan (\frac{1}{3} (sin (x) - 1))) - \frac{1}{3} ln sin^{3} (x) + 8 : \frac{1}{12} ln ∣ sin (x) + 2 ∣ + \frac{1}{24} (- ln \frac{1}{3} (sin (x) - 1)^{2} + 1 + 23 arctan (\frac{1}{3} (sin (x) - 1))) - \frac{1}{3} ln sin^{3} (x) + 8

= \frac{1}{12} ln ∣ sin (x) + 2 ∣ + \frac{1}{24} (- ln \frac{1}{3} (sin (x) - 1)^{2} + 1 + 23 arctan (\frac{1}{3} (sin (x) - 1))) - \frac{1}{3} ln sin^{3} (x) + 8

Add a constant to the solution

= \frac{1}{12} ln ∣ sin (x) + 2 ∣ + \frac{1}{24} (- ln \frac{1}{3} (sin (x) - 1)^{2} + 1 + 23 arctan (\frac{1}{3} (sin (x) - 1))) - \frac{1}{3} ln sin^{3} (x) + 8 + C

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What is the integral of (cos^3(x))/(sin^3(x)+8) ?
The integral of (cos^3(x))/(sin^3(x)+8) is