3.73 \(\int \frac {\text {sech}^2(x)}{a+a \text {sech}(x)} \, dx\)

Optimal. Leaf size=20 \[ \frac {\tan ^{-1}(\sinh (x))}{a}-\frac {\tanh (x)}{a \text {sech}(x)+a} \]

[Out]

arctan(sinh(x))/a-tanh(x)/(a+a*sech(x))

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Rubi [A]  time = 0.07, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3789, 3770, 3794} \[ \frac {\tan ^{-1}(\sinh (x))}{a}-\frac {\tanh (x)}{a \text {sech}(x)+a} \]

Antiderivative was successfully verified.

[In]

Int[Sech[x]^2/(a + a*Sech[x]),x]

[Out]

ArcTan[Sinh[x]]/a - Tanh[x]/(a + a*Sech[x])

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3789

Int[csc[(e_.) + (f_.)*(x_)]^2/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[1/b, Int[Csc[e + f*x],
x], x] - Dist[a/b, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, 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}^2(x)}{a+a \text {sech}(x)} \, dx &=\frac {\int \text {sech}(x) \, dx}{a}-\int \frac {\text {sech}(x)}{a+a \text {sech}(x)} \, dx\\ &=\frac {\tan ^{-1}(\sinh (x))}{a}-\frac {\tanh (x)}{a+a \text {sech}(x)}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 22, normalized size = 1.10 \[ \frac {2 \tan ^{-1}\left (\tanh \left (\frac {x}{2}\right )\right )-\tanh \left (\frac {x}{2}\right )}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[x]^2/(a + a*Sech[x]),x]

[Out]

(2*ArcTan[Tanh[x/2]] - Tanh[x/2])/a

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fricas [A]  time = 0.39, size = 29, normalized size = 1.45 \[ \frac {2 \, {\left ({\left (\cosh \relax (x) + \sinh \relax (x) + 1\right )} \arctan \left (\cosh \relax (x) + \sinh \relax (x)\right ) + 1\right )}}{a \cosh \relax (x) + a \sinh \relax (x) + a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^2/(a+a*sech(x)),x, algorithm="fricas")

[Out]

2*((cosh(x) + sinh(x) + 1)*arctan(cosh(x) + sinh(x)) + 1)/(a*cosh(x) + a*sinh(x) + a)

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giac [A]  time = 0.13, size = 20, normalized size = 1.00 \[ \frac {2 \, \arctan \left (e^{x}\right )}{a} + \frac {2}{a {\left (e^{x} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^2/(a+a*sech(x)),x, algorithm="giac")

[Out]

2*arctan(e^x)/a + 2/(a*(e^x + 1))

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maple [A]  time = 0.08, size = 21, normalized size = 1.05 \[ -\frac {\tanh \left (\frac {x}{2}\right )}{a}+\frac {2 \arctan \left (\tanh \left (\frac {x}{2}\right )\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(x)^2/(a+a*sech(x)),x)

[Out]

-1/a*tanh(1/2*x)+2/a*arctan(tanh(1/2*x))

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maxima [A]  time = 0.45, size = 23, normalized size = 1.15 \[ -\frac {2 \, \arctan \left (e^{\left (-x\right )}\right )}{a} - \frac {2}{a e^{\left (-x\right )} + a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^2/(a+a*sech(x)),x, algorithm="maxima")

[Out]

-2*arctan(e^(-x))/a - 2/(a*e^(-x) + a)

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mupad [B]  time = 1.30, size = 31, normalized size = 1.55 \[ \frac {2}{a\,\left ({\mathrm {e}}^x+1\right )}+\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\sqrt {a^2}}{a}\right )}{\sqrt {a^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(x)^2*(a + a/cosh(x))),x)

[Out]

2/(a*(exp(x) + 1)) + (2*atan((exp(x)*(a^2)^(1/2))/a))/(a^2)^(1/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\operatorname {sech}^{2}{\relax (x )}}{\operatorname {sech}{\relax (x )} + 1}\, dx}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)**2/(a+a*sech(x)),x)

[Out]

Integral(sech(x)**2/(sech(x) + 1), x)/a

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