∫ sech5(x) dx [580]

Optimal. Leaf size=26 \[ \frac {3}{8} \tan ^{-1}(\sinh (x))+\frac {3}{8} \text {sech}(x) \tanh (x)+\frac {1}{4} \text {sech}^3(x) \tanh (x) \]

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Rubi [A]
time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3853, 3855} \begin {gather*} \frac {3}{8} \text {ArcTan}(\sinh (x))+\frac {1}{4} \tanh (x) \text {sech}^3(x)+\frac {3}{8} \tanh (x) \text {sech}(x) \end {gather*}

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,

\begin {align*} \int \text {sech}^5(x) \, dx &=\frac {1}{4} \text {sech}^3(x) \tanh (x)+\frac {3}{4} \int \text {sech}^3(x) \, dx\\ &=\frac {3}{8} \text {sech}(x) \tanh (x)+\frac {1}{4} \text {sech}^3(x) \tanh (x)+\frac {3}{8} \int \text {sech}(x) \, dx\\ &=\frac {3}{8} \tan ^{-1}(\sinh (x))+\frac {3}{8} \text {sech}(x) \tanh (x)+\frac {1}{4} \text {sech}^3(x) \tanh (x)\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 30, normalized size = 1.15 \begin {gather*} \frac {3}{4} \tan ^{-1}\left (\tanh \left (\frac {x}{2}\right )\right )+\frac {3}{8} \text {sech}(x) \tanh (x)+\frac {1}{4} \text {sech}^3(x) \tanh (x) \end {gather*}

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method	result	size

default	\(\left (\frac {\mathrm {sech}\left (x \right )^{3}}{4}+\frac {3 \,\mathrm {sech}\left (x \right )}{8}\right ) \tanh \left (x \right )+\frac {3 \arctan \left ({\mathrm e}^{x}\right )}{4}\)	\(21\)
risch	\(\frac {{\mathrm e}^{x} \left (3 \,{\mathrm e}^{6 x}+11 \,{\mathrm e}^{4 x}-11 \,{\mathrm e}^{2 x}-3\right )}{4 \left (1+{\mathrm e}^{2 x}\right )^{4}}+\frac {3 i \ln \left ({\mathrm e}^{x}+i\right )}{8}-\frac {3 i \ln \left ({\mathrm e}^{x}-i\right )}{8}\)	\(52\)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (20) = 40\).
time = 4.60, size = 61, normalized size = 2.35 \begin {gather*} \frac {3 \, e^{\left (-x\right )} + 11 \, e^{\left (-3 \, x\right )} - 11 \, e^{\left (-5 \, x\right )} - 3 \, e^{\left (-7 \, x\right )}}{4 \, {\left (4 \, e^{\left (-2 \, x\right )} + 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} + e^{\left (-8 \, x\right )} + 1\right )}} - \frac {3}{4} \, \arctan \left (e^{\left (-x\right )}\right ) \end {gather*}

1/4*(3*e^(-x) + 11*e^(-3*x) - 11*e^(-5*x) - 3*e^(-7*x))/(4*e^(-2*x) + 6*e^(-4*x) + 4*e^(-6*x) + e^(-8*x) + 1)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 461 vs. \(2 (20) = 40\).
time = 1.26, size = 461, normalized size = 17.73 \begin {gather*} \frac {3 \, \cosh \left (x\right )^{7} + 21 \, \cosh \left (x\right ) \sinh \left (x\right )^{6} + 3 \, \sinh \left (x\right )^{7} + {\left (63 \, \cosh \left (x\right )^{2} + 11\right )} \sinh \left (x\right )^{5} + 11 \, \cosh \left (x\right )^{5} + 5 \, {\left (21 \, \cosh \left (x\right )^{3} + 11 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{4} + {\left (105 \, \cosh \left (x\right )^{4} + 110 \, \cosh \left (x\right )^{2} - 11\right )} \sinh \left (x\right )^{3} - 11 \, \cosh \left (x\right )^{3} + {\left (63 \, \cosh \left (x\right )^{5} + 110 \, \cosh \left (x\right )^{3} - 33 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + 3 \, {\left (\cosh \left (x\right )^{8} + 8 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sinh \left (x\right )^{8} + 4 \, {\left (7 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{6} + 4 \, \cosh \left (x\right )^{6} + 8 \, {\left (7 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 2 \, {\left (35 \, \cosh \left (x\right )^{4} + 30 \, \cosh \left (x\right )^{2} + 3\right )} \sinh \left (x\right )^{4} + 6 \, \cosh \left (x\right )^{4} + 8 \, {\left (7 \, \cosh \left (x\right )^{5} + 10 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \, {\left (7 \, \cosh \left (x\right )^{6} + 15 \, \cosh \left (x\right )^{4} + 9 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right )^{2} + 8 \, {\left (\cosh \left (x\right )^{7} + 3 \, \cosh \left (x\right )^{5} + 3 \, \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 ) + {\left (21 \, \cosh \left (x\right )^{6} + 55 \, \cosh \left (x\right )^{4} - 33 \, \cosh \left (x\right )^{2} - 3\right )} \sinh \left (x\right ) - 3 \, \cosh \left (x\right )}{4 \, {\left (\cosh \left (x\right )^{8} + 8 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sinh \left (x\right )^{8} + 4 \, {\left (7 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{6} + 4 \, \cosh \left (x\right )^{6} + 8 \, {\left (7 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 2 \, {\left (35 \, \cosh \left (x\right )^{4} + 30 \, \cosh \left (x\right )^{2} + 3\right )} \sinh \left (x\right )^{4} + 6 \, \cosh \left (x\right )^{4} + 8 \, {\left (7 \, \cosh \left (x\right )^{5} + 10 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \, {\left (7 \, \cosh \left (x\right )^{6} + 15 \, \cosh \left (x\right )^{4} + 9 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right )^{2} + 8 \, {\left (\cosh \left (x\right )^{7} + 3 \, \cosh \left (x\right )^{5} + 3 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )}} \end {gather*}

1/4*(3*cosh(x)^7 + 21*cosh(x)*sinh(x)^6 + 3*sinh(x)^7 + (63*cosh(x)^2 + 11)*sinh(x)^5 + 11*cosh(x)^5 + 5*(21*c

osh(x)^3 + 11*cosh(x))*sinh(x)^4 + (105*cosh(x)^4 + 110*cosh(x)^2 - 11)*sinh(x)^3 - 11*cosh(x)^3 + (63*cosh(x)

^5 + 110*cosh(x)^3 - 33*cosh(x))*sinh(x)^2 + 3*(cosh(x)^8 + 8*cosh(x)*sinh(x)^7 + sinh(x)^8 + 4*(7*cosh(x)^2 +

1)*sinh(x)^6 + 4*cosh(x)^6 + 8*(7*cosh(x)^3 + 3*cosh(x))*sinh(x)^5 + 2*(35*cosh(x)^4 + 30*cosh(x)^2 + 3)*sinh

(x)^4 + 6*cosh(x)^4 + 8*(7*cosh(x)^5 + 10*cosh(x)^3 + 3*cosh(x))*sinh(x)^3 + 4*(7*cosh(x)^6 + 15*cosh(x)^4 + 9

*cosh(x)^2 + 1)*sinh(x)^2 + 4*cosh(x)^2 + 8*(cosh(x)^7 + 3*cosh(x)^5 + 3*cosh(x)^3 + cosh(x))*sinh(x) + 1)*arc

tan(cosh(x) + sinh(x)) + (21*cosh(x)^6 + 55*cosh(x)^4 - 33*cosh(x)^2 - 3)*sinh(x) - 3*cosh(x))/(cosh(x)^8 + 8*

cosh(x)*sinh(x)^7 + sinh(x)^8 + 4*(7*cosh(x)^2 + 1)*sinh(x)^6 + 4*cosh(x)^6 + 8*(7*cosh(x)^3 + 3*cosh(x))*sinh

(x)^5 + 2*(35*cosh(x)^4 + 30*cosh(x)^2 + 3)*sinh(x)^4 + 6*cosh(x)^4 + 8*(7*cosh(x)^5 + 10*cosh(x)^3 + 3*cosh(x

))*sinh(x)^3 + 4*(7*cosh(x)^6 + 15*cosh(x)^4 + 9*cosh(x)^2 + 1)*sinh(x)^2 + 4*cosh(x)^2 + 8*(cosh(x)^7 + 3*cos

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 422 vs. \(2 (27) = 54\).
time = 1.25, size = 422, normalized size = 16.23 \begin {gather*} \frac {3 \tanh ^{8}{\left (\frac {x}{2} \right )} \operatorname {atan}{\left (\tanh {\left (\frac {x}{2} \right )} \right )}}{4 \tanh ^{8}{\left (\frac {x}{2} \right )} + 16 \tanh ^{6}{\left (\frac {x}{2} \right )} + 24 \tanh ^{4}{\left (\frac {x}{2} \right )} + 16 \tanh ^{2}{\left (\frac {x}{2} \right )} + 4} - \frac {5 \tanh ^{7}{\left (\frac {x}{2} \right )}}{4 \tanh ^{8}{\left (\frac {x}{2} \right )} + 16 \tanh ^{6}{\left (\frac {x}{2} \right )} + 24 \tanh ^{4}{\left (\frac {x}{2} \right )} + 16 \tanh ^{2}{\left (\frac {x}{2} \right )} + 4} + \frac {12 \tanh ^{6}{\left (\frac {x}{2} \right )} \operatorname {atan}{\left (\tanh {\left (\frac {x}{2} \right )} \right )}}{4 \tanh ^{8}{\left (\frac {x}{2} \right )} + 16 \tanh ^{6}{\left (\frac {x}{2} \right )} + 24 \tanh ^{4}{\left (\frac {x}{2} \right )} + 16 \tanh ^{2}{\left (\frac {x}{2} \right )} + 4} + \frac {3 \tanh ^{5}{\left (\frac {x}{2} \right )}}{4 \tanh ^{8}{\left (\frac {x}{2} \right )} + 16 \tanh ^{6}{\left (\frac {x}{2} \right )} + 24 \tanh ^{4}{\left (\frac {x}{2} \right )} + 16 \tanh ^{2}{\left (\frac {x}{2} \right )} + 4} + \frac {18 \tanh ^{4}{\left (\frac {x}{2} \right )} \operatorname {atan}{\left (\tanh {\left (\frac {x}{2} \right )} \right )}}{4 \tanh ^{8}{\left (\frac {x}{2} \right )} + 16 \tanh ^{6}{\left (\frac {x}{2} \right )} + 24 \tanh ^{4}{\left (\frac {x}{2} \right )} + 16 \tanh ^{2}{\left (\frac {x}{2} \right )} + 4} - \frac {3 \tanh ^{3}{\left (\frac {x}{2} \right )}}{4 \tanh ^{8}{\left (\frac {x}{2} \right )} + 16 \tanh ^{6}{\left (\frac {x}{2} \right )} + 24 \tanh ^{4}{\left (\frac {x}{2} \right )} + 16 \tanh ^{2}{\left (\frac {x}{2} \right )} + 4} + \frac {12 \tanh ^{2}{\left (\frac {x}{2} \right )} \operatorname {atan}{\left (\tanh {\left (\frac {x}{2} \right )} \right )}}{4 \tanh ^{8}{\left (\frac {x}{2} \right )} + 16 \tanh ^{6}{\left (\frac {x}{2} \right )} + 24 \tanh ^{4}{\left (\frac {x}{2} \right )} + 16 \tanh ^{2}{\left (\frac {x}{2} \right )} + 4} + \frac {5 \tanh {\left (\frac {x}{2} \right )}}{4 \tanh ^{8}{\left (\frac {x}{2} \right )} + 16 \tanh ^{6}{\left (\frac {x}{2} \right )} + 24 \tanh ^{4}{\left (\frac {x}{2} \right )} + 16 \tanh ^{2}{\left (\frac {x}{2} \right )} + 4} + \frac {3 \operatorname {atan}{\left (\tanh {\left (\frac {x}{2} \right )} \right )}}{4 \tanh ^{8}{\left (\frac {x}{2} \right )} + 16 \tanh ^{6}{\left (\frac {x}{2} \right )} + 24 \tanh ^{4}{\left (\frac {x}{2} \right )} + 16 \tanh ^{2}{\left (\frac {x}{2} \right )} + 4} \end {gather*}

3*tanh(x/2)**8*atan(tanh(x/2))/(4*tanh(x/2)**8 + 16*tanh(x/2)**6 + 24*tanh(x/2)**4 + 16*tanh(x/2)**2 + 4) - 5*

tanh(x/2)**7/(4*tanh(x/2)**8 + 16*tanh(x/2)**6 + 24*tanh(x/2)**4 + 16*tanh(x/2)**2 + 4) + 12*tanh(x/2)**6*atan

(tanh(x/2))/(4*tanh(x/2)**8 + 16*tanh(x/2)**6 + 24*tanh(x/2)**4 + 16*tanh(x/2)**2 + 4) + 3*tanh(x/2)**5/(4*tan

h(x/2)**8 + 16*tanh(x/2)**6 + 24*tanh(x/2)**4 + 16*tanh(x/2)**2 + 4) + 18*tanh(x/2)**4*atan(tanh(x/2))/(4*tanh

(x/2)**8 + 16*tanh(x/2)**6 + 24*tanh(x/2)**4 + 16*tanh(x/2)**2 + 4) - 3*tanh(x/2)**3/(4*tanh(x/2)**8 + 16*tanh

(x/2)**6 + 24*tanh(x/2)**4 + 16*tanh(x/2)**2 + 4) + 12*tanh(x/2)**2*atan(tanh(x/2))/(4*tanh(x/2)**8 + 16*tanh(

x/2)**6 + 24*tanh(x/2)**4 + 16*tanh(x/2)**2 + 4) + 5*tanh(x/2)/(4*tanh(x/2)**8 + 16*tanh(x/2)**6 + 24*tanh(x/2

)**4 + 16*tanh(x/2)**2 + 4) + 3*atan(tanh(x/2))/(4*tanh(x/2)**8 + 16*tanh(x/2)**6 + 24*tanh(x/2)**4 + 16*tanh(

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (20) = 40\).
time = 0.75, size = 60, normalized size = 2.31 \begin {gather*} \frac {3}{16} \, \pi - \frac {3 \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} + 20 \, e^{\left (-x\right )} - 20 \, e^{x}}{4 \, {\left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4\right )}^{2}} + \frac {3}{8} \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right ) \end {gather*}

3/16*pi - 1/4*(3*(e^(-x) - e^x)^3 + 20*e^(-x) - 20*e^x)/((e^(-x) - e^x)^2 + 4)^2 + 3/8*arctan(1/2*(e^(2*x) - 1

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Mupad [B]
time = 0.08, size = 22, normalized size = 0.85 \begin {gather*} \frac {3\,\mathrm {atan}\left ({\mathrm {e}}^x\right )}{4}+\frac {3\,\mathrm {sinh}\left (x\right )}{8\,{\mathrm {cosh}\left (x\right )}^2}+\frac {\mathrm {sinh}\left (x\right )}{4\,{\mathrm {cosh}\left (x\right )}^4} \end {gather*}

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3.6.80 \(\int \text {sech}^5(x) \, dx\) [580]