\(\int \frac {\text {sech}(x)}{a+a \text {sech}(x)} \, dx\) [72]

Optimal result

Integrand size = 11, antiderivative size = 11 \[ \int \frac {\text {sech}(x)}{a+a \text {sech}(x)} \, dx=\frac {\tanh (x)}{a+a \text {sech}(x)} \]

[Out]

tanh(x)/(a+a*sech(x))

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3879} \[ \int \frac {\text {sech}(x)}{a+a \text {sech}(x)} \, dx=\frac {\tanh (x)}{a \text {sech}(x)+a} \]

[In]

Int[Sech[x]/(a + a*Sech[x]),x]

[Out]

Tanh[x]/(a + a*Sech[x])

Rule 3879

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[-Cot[e + f*x]/(f*(b + a*
Csc[e + f*x])), x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\tanh (x)}{a+a \text {sech}(x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.91 \[ \int \frac {\text {sech}(x)}{a+a \text {sech}(x)} \, dx=\frac {\tanh \left (\frac {x}{2}\right )}{a} \]

[In]

Integrate[Sech[x]/(a + a*Sech[x]),x]

[Out]

Tanh[x/2]/a

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.82

method	result	size
default	\(\frac {\tanh \left (\frac {x}{2}\right )}{a}\)	\(9\)
parallelrisch	\(\frac {\tanh \left (\frac {x}{2}\right )}{a}\)	\(9\)
risch	\(-\frac {2}{\left ({\mathrm e}^{x}+1\right ) a}\)	\(12\)

[In]

int(sech(x)/(a+a*sech(x)),x,method=_RETURNVERBOSE)

[Out]

1/a*tanh(1/2*x)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.27 \[ \int \frac {\text {sech}(x)}{a+a \text {sech}(x)} \, dx=-\frac {2}{a \cosh \left (x\right ) + a \sinh \left (x\right ) + a} \]

[In]

integrate(sech(x)/(a+a*sech(x)),x, algorithm="fricas")

[Out]

-2/(a*cosh(x) + a*sinh(x) + a)

Sympy [F]

\[ \int \frac {\text {sech}(x)}{a+a \text {sech}(x)} \, dx=\frac {\int \frac {\operatorname {sech}{\left (x \right )}}{\operatorname {sech}{\left (x \right )} + 1}\, dx}{a} \]

[In]

integrate(sech(x)/(a+a*sech(x)),x)

[Out]

Integral(sech(x)/(sech(x) + 1), x)/a

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09 \[ \int \frac {\text {sech}(x)}{a+a \text {sech}(x)} \, dx=\frac {2}{a e^{\left (-x\right )} + a} \]

[In]

integrate(sech(x)/(a+a*sech(x)),x, algorithm="maxima")

[Out]

2/(a*e^(-x) + a)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {\text {sech}(x)}{a+a \text {sech}(x)} \, dx=-\frac {2}{a {\left (e^{x} + 1\right )}} \]

[In]

integrate(sech(x)/(a+a*sech(x)),x, algorithm="giac")

[Out]

-2/(a*(e^x + 1))

Mupad [B] (verification not implemented)

Time = 1.93 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {\text {sech}(x)}{a+a \text {sech}(x)} \, dx=-\frac {2}{a\,\left ({\mathrm {e}}^x+1\right )} \]

[In]

int(1/(cosh(x)*(a + a/cosh(x))),x)

[Out]

-2/(a*(exp(x) + 1))