3.1.95 \(\int \frac {\text {sech}(x)}{1+\tanh (x)} \, dx\) [95]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 10, normalized size of antiderivative = 1.00, 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 = {3569} \begin {gather*} -\frac {\text {sech}(x)}{\tanh (x)+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sech[x]/(1 + Tanh[x]),x]

[Out]

-(Sech[x]/(1 + Tanh[x]))

Rule 3569

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*}

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Mathematica [A]
time = 0.01, size = 7, normalized size = 0.70 \begin {gather*} -\cosh (x)+\sinh (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sech[x]/(1 + Tanh[x]),x]

[Out]

-Cosh[x] + Sinh[x]

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Maple [A]
time = 0.26, size = 11, normalized size = 1.10

method	result	size
risch	\(-{\mathrm e}^{-x}\)	\(7\)
gosper	\(-\frac {\mathrm {sech}\left (x \right )}{1+\tanh \left (x \right )}\)	\(11\)
default	\(-\frac {2}{\tanh \left (\frac {x}{2}\right )+1}\)	\(11\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(x)/(1+tanh(x)),x,method=_RETURNVERBOSE)

[Out]

-2/(tanh(1/2*x)+1)

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Maxima [A]
time = 0.27, size = 6, normalized size = 0.60 \begin {gather*} -e^{\left (-x\right )} \end {gather*}

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 = 0.37, size = 9, normalized size = 0.90 \begin {gather*} -\frac {1}{\cosh \left (x\right ) + \sinh \left (x\right )} \end {gather*}

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 = 0.16, size = 8, normalized size = 0.80 \begin {gather*} - \frac {\operatorname {sech}{\left (x \right )}}{\tanh {\left (x \right )} + 1} \end {gather*}

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 = 0.41, size = 6, normalized size = 0.60 \begin {gather*} -e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e^(-x)

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Mupad [B]
time = 0.04, size = 6, normalized size = 0.60 \begin {gather*} -{\mathrm {e}}^{-x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(x)*(tanh(x) + 1)),x)

[Out]

-exp(-x)

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