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 = 0.02, 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 = {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*}

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Mathematica [A]  time = 0.00, size = 7, normalized size = 0.70 \[ \sinh (x)-\cosh (x) \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[x]/(1 + Tanh[x]),x]

[Out]

-Cosh[x] + Sinh[x]

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fricas [A]  time = 0.47, size = 9, normalized size = 0.90 \[ -\frac {1}{\cosh \relax (x) + \sinh \relax (x)} \]

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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giac [A]  time = 0.11, size = 6, normalized size = 0.60 \[ -e^{\left (-x\right )} \]

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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maple [A]  time = 0.01, size = 11, normalized size = 1.10 \[ -\frac {\mathrm {sech}\relax (x )}{1+\tanh \relax (x )} \]

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 = 0.31, size = 6, normalized size = 0.60 \[ -e^{\left (-x\right )} \]

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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mupad [B]  time = 0.04, size = 6, normalized size = 0.60 \[ -{\mathrm {e}}^{-x} \]

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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sympy [A]  time = 0.30, size = 8, normalized size = 0.80 \[ - \frac {\operatorname {sech}{\relax (x )}}{\tanh {\relax (x )} + 1} \]

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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