3.72 \(\int \frac{\text{sech}(x)}{a+a \text{sech}(x)} \, dx\)

Optimal. Leaf size=11 \[ \frac{\tanh (x)}{a \text{sech}(x)+a} \]

[Out]

Tanh[x]/(a + a*Sech[x])

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Rubi [A]  time = 0.0242042, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {3794} \[ \frac{\tanh (x)}{a \text{sech}(x)+a} \]

Antiderivative was successfully verified.

[In]

Int[Sech[x]/(a + a*Sech[x]),x]

[Out]

Tanh[x]/(a + a*Sech[x])

Rule 3794

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

Mathematica [A]  time = 0.0073096, size = 10, normalized size = 0.91 \[ \frac{\tanh \left (\frac{x}{2}\right )}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[x]/(a + a*Sech[x]),x]

[Out]

Tanh[x/2]/a

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Maple [A]  time = 0.012, size = 9, normalized size = 0.8 \begin{align*}{\frac{1}{a}\tanh \left ({\frac{x}{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.13703, size = 16, normalized size = 1.45 \begin{align*} \frac{2}{a e^{\left (-x\right )} + a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(a*e^(-x) + a)

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Fricas [A]  time = 2.27118, size = 43, normalized size = 3.91 \begin{align*} -\frac{2}{a \cosh \left (x\right ) + a \sinh \left (x\right ) + a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(a*cosh(x) + a*sinh(x) + a)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\operatorname{sech}{\left (x \right )}}{\operatorname{sech}{\left (x \right )} + 1}\, dx}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sech(x)/(sech(x) + 1), x)/a

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Giac [A]  time = 1.14796, size = 15, normalized size = 1.36 \begin{align*} -\frac{2}{a{\left (e^{x} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(a*(e^x + 1))