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 = 0.02, antiderivative size = 11, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 0.37, size = 14, normalized size = 1.27 \[ -\frac {2}{a \cosh \relax (x) + a \sinh \relax (x) + a} \]

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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giac [A]  time = 0.14, size = 11, normalized size = 1.00 \[ -\frac {2}{a {\left (e^{x} + 1\right )}} \]

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))

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maple [A]  time = 0.07, size = 9, normalized size = 0.82 \[ \frac {\tanh \left (\frac {x}{2}\right )}{a} \]

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 = 0.33, size = 12, normalized size = 1.09 \[ \frac {2}{a e^{\left (-x\right )} + a} \]

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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mupad [B]  time = 1.31, size = 11, normalized size = 1.00 \[ -\frac {2}{a\,\left ({\mathrm {e}}^x+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(x)*(a + a/cosh(x))),x)

[Out]

-2/(a*(exp(x) + 1))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\operatorname {sech}{\relax (x )}}{\operatorname {sech}{\relax (x )} + 1}\, dx}{a} \]

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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