3.99 \(\int \frac{\text{sech}^5(x)}{1+\tanh (x)} \, dx\)

Optimal. Leaf size=24 \[ \frac{\text{sech}^3(x)}{3}+\frac{1}{2} \tan ^{-1}(\sinh (x))+\frac{1}{2} \tanh (x) \text{sech}(x) \]

[Out]

ArcTan[Sinh[x]]/2 + Sech[x]^3/3 + (Sech[x]*Tanh[x])/2

________________________________________________________________________________________

Rubi [A]  time = 0.0422554, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {3501, 3768, 3770} \[ \frac{\text{sech}^3(x)}{3}+\frac{1}{2} \tan ^{-1}(\sinh (x))+\frac{1}{2} \tanh (x) \text{sech}(x) \]

Antiderivative was successfully verified.

[In]

Int[Sech[x]^5/(1 + Tanh[x]),x]

[Out]

ArcTan[Sinh[x]]/2 + Sech[x]^3/3 + (Sech[x]*Tanh[x])/2

Rule 3501

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d^2*
(d*Sec[e + f*x])^(m - 2)*(a + b*Tan[e + f*x])^(n + 1))/(b*f*(m + n - 1)), x] + Dist[(d^2*(m - 2))/(a*(m + n -
1)), Int[(d*Sec[e + f*x])^(m - 2)*(a + b*Tan[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2
+ b^2, 0] && LtQ[n, 0] && GtQ[m, 1] &&  !ILtQ[m + n, 0] && NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3770

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

Rubi steps

\begin{align*} \int \frac{\text{sech}^5(x)}{1+\tanh (x)} \, dx &=\frac{\text{sech}^3(x)}{3}+\int \text{sech}^3(x) \, dx\\ &=\frac{\text{sech}^3(x)}{3}+\frac{1}{2} \text{sech}(x) \tanh (x)+\frac{1}{2} \int \text{sech}(x) \, dx\\ &=\frac{1}{2} \tan ^{-1}(\sinh (x))+\frac{\text{sech}^3(x)}{3}+\frac{1}{2} \text{sech}(x) \tanh (x)\\ \end{align*}

Mathematica [A]  time = 0.0270483, size = 24, normalized size = 1. \[ \frac{\text{sech}^3(x)}{3}+\tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right )+\frac{1}{2} \tanh (x) \text{sech}(x) \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[x]^5/(1 + Tanh[x]),x]

[Out]

ArcTan[Tanh[x/2]] + Sech[x]^3/3 + (Sech[x]*Tanh[x])/2

________________________________________________________________________________________

Maple [B]  time = 0.026, size = 41, normalized size = 1.7 \begin{align*} 2\,{\frac{-1/2\, \left ( \tanh \left ( x/2 \right ) \right ) ^{5}+ \left ( \tanh \left ( x/2 \right ) \right ) ^{4}+1/2\,\tanh \left ( x/2 \right ) +1/3}{ \left ( \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{3}}}+\arctan \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(x)^5/(1+tanh(x)),x)

[Out]

2*(-1/2*tanh(1/2*x)^5+tanh(1/2*x)^4+1/2*tanh(1/2*x)+1/3)/(tanh(1/2*x)^2+1)^3+arctan(tanh(1/2*x))

________________________________________________________________________________________

Maxima [B]  time = 1.63739, size = 66, normalized size = 2.75 \begin{align*} \frac{3 \, e^{\left (-x\right )} + 8 \, e^{\left (-3 \, x\right )} - 3 \, e^{\left (-5 \, x\right )}}{3 \,{\left (3 \, e^{\left (-2 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} + 1\right )}} - \arctan \left (e^{\left (-x\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^5/(1+tanh(x)),x, algorithm="maxima")

[Out]

1/3*(3*e^(-x) + 8*e^(-3*x) - 3*e^(-5*x))/(3*e^(-2*x) + 3*e^(-4*x) + e^(-6*x) + 1) - arctan(e^(-x))

________________________________________________________________________________________

Fricas [B]  time = 2.21472, size = 968, normalized size = 40.33 \begin{align*} \frac{3 \, \cosh \left (x\right )^{5} + 15 \, \cosh \left (x\right ) \sinh \left (x\right )^{4} + 3 \, \sinh \left (x\right )^{5} + 2 \,{\left (15 \, \cosh \left (x\right )^{2} + 4\right )} \sinh \left (x\right )^{3} + 8 \, \cosh \left (x\right )^{3} + 6 \,{\left (5 \, \cosh \left (x\right )^{3} + 4 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + 3 \,{\left (\cosh \left (x\right )^{6} + 6 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6} + 3 \,{\left (5 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{4} + 3 \, \cosh \left (x\right )^{4} + 4 \,{\left (5 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \,{\left (5 \, \cosh \left (x\right )^{4} + 6 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 3 \, \cosh \left (x\right )^{2} + 6 \,{\left (\cosh \left (x\right )^{5} + 2 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) + 3 \,{\left (5 \, \cosh \left (x\right )^{4} + 8 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right ) - 3 \, \cosh \left (x\right )}{3 \,{\left (\cosh \left (x\right )^{6} + 6 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6} + 3 \,{\left (5 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{4} + 3 \, \cosh \left (x\right )^{4} + 4 \,{\left (5 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \,{\left (5 \, \cosh \left (x\right )^{4} + 6 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 3 \, \cosh \left (x\right )^{2} + 6 \,{\left (\cosh \left (x\right )^{5} + 2 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^5/(1+tanh(x)),x, algorithm="fricas")

[Out]

1/3*(3*cosh(x)^5 + 15*cosh(x)*sinh(x)^4 + 3*sinh(x)^5 + 2*(15*cosh(x)^2 + 4)*sinh(x)^3 + 8*cosh(x)^3 + 6*(5*co
sh(x)^3 + 4*cosh(x))*sinh(x)^2 + 3*(cosh(x)^6 + 6*cosh(x)*sinh(x)^5 + sinh(x)^6 + 3*(5*cosh(x)^2 + 1)*sinh(x)^
4 + 3*cosh(x)^4 + 4*(5*cosh(x)^3 + 3*cosh(x))*sinh(x)^3 + 3*(5*cosh(x)^4 + 6*cosh(x)^2 + 1)*sinh(x)^2 + 3*cosh
(x)^2 + 6*(cosh(x)^5 + 2*cosh(x)^3 + cosh(x))*sinh(x) + 1)*arctan(cosh(x) + sinh(x)) + 3*(5*cosh(x)^4 + 8*cosh
(x)^2 - 1)*sinh(x) - 3*cosh(x))/(cosh(x)^6 + 6*cosh(x)*sinh(x)^5 + sinh(x)^6 + 3*(5*cosh(x)^2 + 1)*sinh(x)^4 +
3*cosh(x)^4 + 4*(5*cosh(x)^3 + 3*cosh(x))*sinh(x)^3 + 3*(5*cosh(x)^4 + 6*cosh(x)^2 + 1)*sinh(x)^2 + 3*cosh(x)
^2 + 6*(cosh(x)^5 + 2*cosh(x)^3 + cosh(x))*sinh(x) + 1)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{sech}^{5}{\left (x \right )}}{\tanh{\left (x \right )} + 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)**5/(1+tanh(x)),x)

[Out]

Integral(sech(x)**5/(tanh(x) + 1), x)

________________________________________________________________________________________

Giac [A]  time = 1.22936, size = 42, normalized size = 1.75 \begin{align*} \frac{3 \, e^{\left (5 \, x\right )} + 8 \, e^{\left (3 \, x\right )} - 3 \, e^{x}}{3 \,{\left (e^{\left (2 \, x\right )} + 1\right )}^{3}} + \arctan \left (e^{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^5/(1+tanh(x)),x, algorithm="giac")

[Out]

1/3*(3*e^(5*x) + 8*e^(3*x) - 3*e^x)/(e^(2*x) + 1)^3 + arctan(e^x)