3.95 \(\int \frac{\text{sech}(x)}{1+\tanh (x)} \, dx\)

Optimal. Leaf size=10 \[ -\frac{\text{sech}(x)}{\tanh (x)+1} \]

[Out]

-(Sech[x]/(1 + Tanh[x]))

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Rubi [A]  time = 0.019921, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {3488} \[ -\frac{\text{sech}(x)}{\tanh (x)+1} \]

Antiderivative was successfully verified.

[In]

Int[Sech[x]/(1 + Tanh[x]),x]

[Out]

-(Sech[x]/(1 + Tanh[x]))

Rule 3488

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(d
*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^n)/(a*f*m), x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0] &
& EqQ[Simplify[m + n], 0]

Rubi steps

\begin{align*} \int \frac{\text{sech}(x)}{1+\tanh (x)} \, dx &=-\frac{\text{sech}(x)}{1+\tanh (x)}\\ \end{align*}

Mathematica [A]  time = 0.0032273, size = 7, normalized size = 0.7 \[ \sinh (x)-\cosh (x) \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[x]/(1 + Tanh[x]),x]

[Out]

-Cosh[x] + Sinh[x]

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Maple [A]  time = 0.003, size = 11, normalized size = 1.1 \begin{align*} -{\frac{{\rm sech} \left (x\right )}{1+\tanh \left ( x \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(x)/(1+tanh(x)),x)

[Out]

-sech(x)/(1+tanh(x))

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Maxima [A]  time = 1.17869, size = 8, normalized size = 0.8 \begin{align*} -e^{\left (-x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)/(1+tanh(x)),x, algorithm="maxima")

[Out]

-e^(-x)

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Fricas [A]  time = 2.23128, size = 32, normalized size = 3.2 \begin{align*} -\frac{1}{\cosh \left (x\right ) + \sinh \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)/(1+tanh(x)),x, algorithm="fricas")

[Out]

-1/(cosh(x) + sinh(x))

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Sympy [A]  time = 0.381505, size = 8, normalized size = 0.8 \begin{align*} - \frac{\operatorname{sech}{\left (x \right )}}{\tanh{\left (x \right )} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)/(1+tanh(x)),x)

[Out]

-sech(x)/(tanh(x) + 1)

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Giac [A]  time = 1.19, size = 8, normalized size = 0.8 \begin{align*} -e^{\left (-x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)/(1+tanh(x)),x, algorithm="giac")

[Out]

-e^(-x)