What is the integral of (sech^2(x))/(tanh(x)) ?

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

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integral of (sech^2(x))/(tanh(x))

\int \frac{sech ^{2} ( x )}{tanh ( x )} d x

ln ∣ sinh (x) ∣ - ln ∣ cosh (x) ∣ + C

Solution steps

\int \frac{sech ^{2} ( x )}{tanh ( x )} d x

Rewrite using trig identities

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

Expand

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

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

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

\int tanh (x) d x = ln ∣ cosh (x) ∣

= ln ∣ sinh (x) ∣ - ln ∣ cosh (x) ∣

Add a constant to the solution

= ln ∣ sinh (x) ∣ - ln ∣ cosh (x) ∣ + C

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