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 = 0.10, antiderivative size = 26, normalized size of antiderivative = 1.00, 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 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 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 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*}

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Mathematica [A]  time = 0.09, 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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fricas [B]  time = 0.38, size = 127, normalized size = 4.88 \[ -\frac {2 \, {\left ({\left (\cosh \relax (x)^{3} + {\left (3 \, \cosh \relax (x) + 1\right )} \sinh \relax (x)^{2} + \sinh \relax (x)^{3} + \cosh \relax (x)^{2} + {\left (3 \, \cosh \relax (x)^{2} + 2 \, \cosh \relax (x) + 1\right )} \sinh \relax (x) + \cosh \relax (x) + 1\right )} \arctan \left (\cosh \relax (x) + \sinh \relax (x)\right ) + \cosh \relax (x)^{2} + {\left (2 \, \cosh \relax (x) + 1\right )} \sinh \relax (x) + \sinh \relax (x)^{2} + \cosh \relax (x) + 2\right )}}{a \cosh \relax (x)^{3} + a \sinh \relax (x)^{3} + a \cosh \relax (x)^{2} + {\left (3 \, a \cosh \relax (x) + a\right )} \sinh \relax (x)^{2} + a \cosh \relax (x) + {\left (3 \, a \cosh \relax (x)^{2} + 2 \, a \cosh \relax (x) + a\right )} \sinh \relax (x) + a} \]

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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giac [A]  time = 0.13, size = 36, normalized size = 1.38 \[ -\frac {2 \, \arctan \left (e^{x}\right )}{a} - \frac {2 \, {\left (e^{\left (2 \, x\right )} + e^{x} + 2\right )}}{a {\left (e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )} + e^{x} + 1\right )}} \]

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))

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maple [A]  time = 0.09, size = 39, normalized size = 1.50 \[ \frac {\tanh \left (\frac {x}{2}\right )}{a}+\frac {2 \tanh \left (\frac {x}{2}\right )}{a \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )}-\frac {2 \arctan \left (\tanh \left (\frac {x}{2}\right )\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(x)^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 = 0.42, size = 45, normalized size = 1.73 \[ \frac {2 \, {\left (e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 2\right )}}{a e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} + a e^{\left (-3 \, x\right )} + a} + \frac {2 \, \arctan \left (e^{\left (-x\right )}\right )}{a} \]

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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mupad [B]  time = 1.32, size = 58, normalized size = 2.23 \[ -\frac {\frac {2\,{\mathrm {e}}^{2\,x}}{a}+\frac {4}{a}+\frac {2\,{\mathrm {e}}^x}{a}}{{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{3\,x}+{\mathrm {e}}^x+1}-\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\sqrt {a^2}}{a}\right )}{\sqrt {a^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(x)^3*(a + a/cosh(x))),x)

[Out]

- ((2*exp(2*x))/a + 4/a + (2*exp(x))/a)/(exp(2*x) + exp(3*x) + 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 \[ \frac {\int \frac {\operatorname {sech}^{3}{\relax (x )}}{\operatorname {sech}{\relax (x )} + 1}\, dx}{a} \]

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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