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

The integral of tanh^3(x) is -ln|sech(x)|+(sech^2(x))/2+C

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integral of tanh^3(x)

\int tanh^{3} (x) d x

- ln ∣ sech (x) ∣ + \frac{sech ^{2} ( x )}{2} + C

Solution steps

\int tanh^{3} (x) d x

Rewrite using trig identities

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

Apply u-substitution

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

Expand

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

= \int - \frac{1}{u} + u 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}{u} d u + \int u d u

\int \frac{1}{u} d u = ln ∣ u ∣

\int u d u = \frac{u ^{2}}{2}

= - ln ∣ u ∣ + \frac{u ^{2}}{2}

Substitute back

u = sech (x)

= - ln ∣ sech (x) ∣ + \frac{sech ^{2} ( x )}{2}

Add a constant to the solution

= - ln ∣ sech (x) ∣ + \frac{sech ^{2} ( x )}{2} + C

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What is the integral of tanh^3(x) ?
The integral of tanh^3(x) is -ln|sech(x)|+(sech^2(x))/2+C