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

Optimal result

Integrand size = 9, antiderivative size = 10 \[ \int \frac {\text {sech}(x)}{1+\tanh (x)} \, dx=-\frac {\text {sech}(x)}{1+\tanh (x)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {3569} \[ \int \frac {\text {sech}(x)}{1+\tanh (x)} \, dx=-\frac {\text {sech}(x)}{\tanh (x)+1} \]

[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*} \text {integral}& = -\frac {\text {sech}(x)}{1+\tanh (x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.70 \[ \int \frac {\text {sech}(x)}{1+\tanh (x)} \, dx=-\cosh (x)+\sinh (x) \]

[In]

Integrate[Sech[x]/(1 + Tanh[x]),x]

[Out]

-Cosh[x] + Sinh[x]

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.70

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

[In]

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

[Out]

-exp(-x)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.90 \[ \int \frac {\text {sech}(x)}{1+\tanh (x)} \, dx=-\frac {1}{\cosh \left (x\right ) + \sinh \left (x\right )} \]

[In]

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

[Out]

-1/(cosh(x) + sinh(x))

Sympy [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {\text {sech}(x)}{1+\tanh (x)} \, dx=- \frac {\operatorname {sech}{\left (x \right )}}{\tanh {\left (x \right )} + 1} \]

[In]

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

[Out]

-sech(x)/(tanh(x) + 1)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.60 \[ \int \frac {\text {sech}(x)}{1+\tanh (x)} \, dx=-e^{\left (-x\right )} \]

[In]

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

[Out]

-e^(-x)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.60 \[ \int \frac {\text {sech}(x)}{1+\tanh (x)} \, dx=-e^{\left (-x\right )} \]

[In]

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

[Out]

-e^(-x)

Mupad [B] (verification not implemented)

Time = 1.65 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.60 \[ \int \frac {\text {sech}(x)}{1+\tanh (x)} \, dx=-{\mathrm {e}}^{-x} \]

[In]

int(1/(cosh(x)*(tanh(x) + 1)),x)

[Out]

-exp(-x)