What is the integral of (1-tanh^3(3x))/(tanh(3x)) ?

The integral of (1-tanh^3(3x))/(tanh(3x)) is 1/3 ln|sinh(3x)|-x+1/3 tanh(3x)+C

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integral of (1-tanh^3(3x))/(tanh(3x))

\int \frac{1 - tanh ^{3} ( 3 x )}{tanh ( 3 x )} d x

\frac{1}{3} ln ∣ sinh (3 x) ∣ - x + \frac{1}{3} tanh (3 x) + C

Solution steps

\int \frac{1 - tanh ^{3} ( 3 x )}{tanh ( 3 x )} d x

Expand

\frac{1 - tanh ^{3} ( 3 x )}{tanh ( 3 x )} : \frac{1}{tanh ( 3 x )} - tanh^{2} (3 x)

= \int \frac{1}{tanh ( 3 x )} - tanh^{2} (3 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}{tanh ( 3 x )} d x - \int tanh^{2} (3 x) d x

\int \frac{1}{tanh ( 3 x )} d x = \frac{1}{3} ln ∣ sinh (3 x) ∣

\int tanh^{2} (3 x) d x = x - \frac{1}{3} tanh (3 x)

= \frac{1}{3} ln ∣ sinh (3 x) ∣ - (x - \frac{1}{3} tanh (3 x))

Simplify

= \frac{1}{3} ln ∣ sinh (3 x) ∣ - x + \frac{1}{3} tanh (3 x)

Add a constant to the solution

= \frac{1}{3} ln ∣ sinh (3 x) ∣ - x + \frac{1}{3} tanh (3 x) + C

\int \frac{4 x ^{3}}{3} d x

0.5 y^{^{''}} + 10 y^{^{'}} + 100 y = 150

x \to 1 lim ((x)^{\frac{1}{1 - x}})

\frac{d}{d x} (- \frac{2 x}{y})

\int \frac{4 x ^{2} - 25}{x} d x

What is the integral of (1-tanh^3(3x))/(tanh(3x)) ?
The integral of (1-tanh^3(3x))/(tanh(3x)) is 1/3 ln|sinh(3x)|-x+1/3 tanh(3x)+C