3.1.99 \(\int \frac {\text {sech}^5(x)}{1+\tanh (x)} \, dx\) [99]

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

[Out]

1/2*arctan(sinh(x))+1/3*sech(x)^3+1/2*sech(x)*tanh(x)

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Rubi [A]
time = 0.03, antiderivative size = 24, normalized size of antiderivative = 1.00, 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 = {3582, 3853, 3855} \begin {gather*} \frac {1}{2} \text {ArcTan}(\sinh (x))+\frac {\text {sech}^3(x)}{3}+\frac {1}{2} \tanh (x) \text {sech}(x) \end {gather*}

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 3582

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 3853

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 3855

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*}

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Mathematica [A]
time = 0.02, size = 24, normalized size = 1.00 \begin {gather*} \text {ArcTan}\left (\tanh \left (\frac {x}{2}\right )\right )+\frac {\text {sech}^3(x)}{3}+\frac {1}{2} \text {sech}(x) \tanh (x) \end {gather*}

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(40\) vs. \(2(18)=36\).
time = 0.57, size = 41, normalized size = 1.71

method	result	size
default	\(\frac {-\left (\tanh ^{5}\left (\frac {x}{2}\right )\right )+2 \left (\tanh ^{4}\left (\frac {x}{2}\right )\right )+\tanh \left (\frac {x}{2}\right )+\frac {2}{3}}{\left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{3}}+\arctan \left (\tanh \left (\frac {x}{2}\right )\right )\)	\(41\)
risch	\(\frac {{\mathrm e}^{x} \left (3 \,{\mathrm e}^{4 x}+8 \,{\mathrm e}^{2 x}-3\right )}{3 \left (1+{\mathrm e}^{2 x}\right )^{3}}+\frac {i \ln \left ({\mathrm e}^{x}+i\right )}{2}-\frac {i \ln \left ({\mathrm e}^{x}-i\right )}{2}\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (18) = 36\).
time = 0.46, size = 49, normalized size = 2.04 \begin {gather*} \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 {gather*}

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))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 288 vs. \(2 (18) = 36\).
time = 0.42, size = 288, normalized size = 12.00 \begin {gather*} \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 {gather*}

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)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {sech}^{5}{\left (x \right )}}{\tanh {\left (x \right )} + 1}\, dx \end {gather*}

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)

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Giac [A]
time = 0.41, size = 31, normalized size = 1.29 \begin {gather*} \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 {gather*}

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)

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Mupad [B]
time = 0.09, size = 61, normalized size = 2.54 \begin {gather*} \mathrm {atan}\left ({\mathrm {e}}^x\right )+\frac {{\mathrm {e}}^x}{{\mathrm {e}}^{2\,x}+1}-\frac {8\,{\mathrm {e}}^x}{3\,\left (3\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}+1\right )}+\frac {2\,{\mathrm {e}}^x}{3\,\left (2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(x)^5*(tanh(x) + 1)),x)

[Out]

atan(exp(x)) + exp(x)/(exp(2*x) + 1) - (8*exp(x))/(3*(3*exp(2*x) + 3*exp(4*x) + exp(6*x) + 1)) + (2*exp(x))/(3
*(2*exp(2*x) + exp(4*x) + 1))

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