3.74 \(\int \frac{\text{sech}^3(x)}{a+a \text{sech}(x)} \, dx\)

Optimal. Leaf size=26 \[ \frac{\tanh (x)}{a}-\frac{\tan ^{-1}(\sinh (x))}{a}+\frac{\tanh (x)}{a \text{sech}(x)+a} \]

[Out]

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

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Rubi [A]  time = 0.101052, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {3790, 3789, 3770, 3794} \[ \frac{\tanh (x)}{a}-\frac{\tan ^{-1}(\sinh (x))}{a}+\frac{\tanh (x)}{a \text{sech}(x)+a} \]

Antiderivative was successfully verified.

[In]

Int[Sech[x]^3/(a + a*Sech[x]),x]

[Out]

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

Rule 3790

Int[csc[(e_.) + (f_.)*(x_)]^3/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> -Simp[Cot[e + f*x]/(b*f), x
] - Dist[a/b, Int[Csc[e + f*x]^2/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, 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 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, 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}^3(x)}{a+a \text{sech}(x)} \, dx &=\frac{\tanh (x)}{a}-\int \frac{\text{sech}^2(x)}{a+a \text{sech}(x)} \, dx\\ &=\frac{\tanh (x)}{a}-\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}+\frac{\tanh (x)}{a+a \text{sech}(x)}\\ \end{align*}

Mathematica [A]  time = 0.0851934, size = 45, normalized size = 1.73 \[ \frac{2 \cosh \left (\frac{x}{2}\right ) \text{sech}(x) \left (\sinh \left (\frac{x}{2}\right )+\cosh \left (\frac{x}{2}\right ) \left (\tanh (x)-2 \tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right )\right )\right )}{a (\text{sech}(x)+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[x]^3/(a + a*Sech[x]),x]

[Out]

(2*Cosh[x/2]*Sech[x]*(Sinh[x/2] + Cosh[x/2]*(-2*ArcTan[Tanh[x/2]] + Tanh[x])))/(a*(1 + Sech[x]))

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Maple [A]  time = 0.015, size = 39, normalized size = 1.5 \begin{align*}{\frac{1}{a}\tanh \left ({\frac{x}{2}} \right ) }+2\,{\frac{\tanh \left ( x/2 \right ) }{a \left ( \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+1 \right ) }}-2\,{\frac{\arctan \left ( \tanh \left ( x/2 \right ) \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(x)^3/(a+a*sech(x)),x)

[Out]

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

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Maxima [A]  time = 1.67018, size = 61, normalized size = 2.35 \begin{align*} \frac{2 \,{\left (e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 2\right )}}{a e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} + a e^{\left (-3 \, x\right )} + a} + \frac{2 \, \arctan \left (e^{\left (-x\right )}\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^3/(a+a*sech(x)),x, algorithm="maxima")

[Out]

2*(e^(-x) + e^(-2*x) + 2)/(a*e^(-x) + a*e^(-2*x) + a*e^(-3*x) + a) + 2*arctan(e^(-x))/a

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Fricas [B]  time = 2.29869, size = 467, normalized size = 17.96 \begin{align*} -\frac{2 \,{\left ({\left (\cosh \left (x\right )^{3} +{\left (3 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3} + \cosh \left (x\right )^{2} +{\left (3 \, \cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + \cosh \left (x\right ) + 1\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) + \cosh \left (x\right )^{2} +{\left (2 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} + \cosh \left (x\right ) + 2\right )}}{a \cosh \left (x\right )^{3} + a \sinh \left (x\right )^{3} + a \cosh \left (x\right )^{2} +{\left (3 \, a \cosh \left (x\right ) + a\right )} \sinh \left (x\right )^{2} + a \cosh \left (x\right ) +{\left (3 \, a \cosh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) + a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^3/(a+a*sech(x)),x, algorithm="fricas")

[Out]

-2*((cosh(x)^3 + (3*cosh(x) + 1)*sinh(x)^2 + sinh(x)^3 + cosh(x)^2 + (3*cosh(x)^2 + 2*cosh(x) + 1)*sinh(x) + c
osh(x) + 1)*arctan(cosh(x) + sinh(x)) + cosh(x)^2 + (2*cosh(x) + 1)*sinh(x) + sinh(x)^2 + cosh(x) + 2)/(a*cosh
(x)^3 + a*sinh(x)^3 + a*cosh(x)^2 + (3*a*cosh(x) + a)*sinh(x)^2 + a*cosh(x) + (3*a*cosh(x)^2 + 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}^{3}{\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)**3/(a+a*sech(x)),x)

[Out]

Integral(sech(x)**3/(sech(x) + 1), x)/a

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Giac [A]  time = 1.11443, size = 49, normalized size = 1.88 \begin{align*} -\frac{2 \, \arctan \left (e^{x}\right )}{a} - \frac{2 \,{\left (e^{\left (2 \, x\right )} + e^{x} + 2\right )}}{a{\left (e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )} + e^{x} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^3/(a+a*sech(x)),x, algorithm="giac")

[Out]

-2*arctan(e^x)/a - 2*(e^(2*x) + e^x + 2)/(a*(e^(3*x) + e^(2*x) + e^x + 1))