Optimal. Leaf size=26 \[ \frac {3}{8} \tan ^{-1}(\sinh (x))+\frac {1}{4} \tanh (x) \text {sech}^3(x)+\frac {3}{8} \tanh (x) \text {sech}(x) \]
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Rubi [A] time = 0.02, 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 = {3768, 3770} \[ \frac {3}{8} \tan ^{-1}(\sinh (x))+\frac {1}{4} \tanh (x) \text {sech}^3(x)+\frac {3}{8} \tanh (x) \text {sech}(x) \]
Antiderivative was successfully verified.
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Rule 3768
Rule 3770
Rubi steps
\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 \[ \frac {3}{4} \tan ^{-1}\left (\tanh \left (\frac {x}{2}\right )\right )+\frac {1}{4} \tanh (x) \text {sech}^3(x)+\frac {3}{8} \tanh (x) \text {sech}(x) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \text {sech}^5(x) \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 1.27, size = 461, normalized size = 17.73 \[ \frac {3 \, \cosh \relax (x)^{7} + 21 \, \cosh \relax (x) \sinh \relax (x)^{6} + 3 \, \sinh \relax (x)^{7} + {\left (63 \, \cosh \relax (x)^{2} + 11\right )} \sinh \relax (x)^{5} + 11 \, \cosh \relax (x)^{5} + 5 \, {\left (21 \, \cosh \relax (x)^{3} + 11 \, \cosh \relax (x)\right )} \sinh \relax (x)^{4} + {\left (105 \, \cosh \relax (x)^{4} + 110 \, \cosh \relax (x)^{2} - 11\right )} \sinh \relax (x)^{3} - 11 \, \cosh \relax (x)^{3} + {\left (63 \, \cosh \relax (x)^{5} + 110 \, \cosh \relax (x)^{3} - 33 \, \cosh \relax (x)\right )} \sinh \relax (x)^{2} + 3 \, {\left (\cosh \relax (x)^{8} + 8 \, \cosh \relax (x) \sinh \relax (x)^{7} + \sinh \relax (x)^{8} + 4 \, {\left (7 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{6} + 4 \, \cosh \relax (x)^{6} + 8 \, {\left (7 \, \cosh \relax (x)^{3} + 3 \, \cosh \relax (x)\right )} \sinh \relax (x)^{5} + 2 \, {\left (35 \, \cosh \relax (x)^{4} + 30 \, \cosh \relax (x)^{2} + 3\right )} \sinh \relax (x)^{4} + 6 \, \cosh \relax (x)^{4} + 8 \, {\left (7 \, \cosh \relax (x)^{5} + 10 \, \cosh \relax (x)^{3} + 3 \, \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 4 \, {\left (7 \, \cosh \relax (x)^{6} + 15 \, \cosh \relax (x)^{4} + 9 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{2} + 4 \, \cosh \relax (x)^{2} + 8 \, {\left (\cosh \relax (x)^{7} + 3 \, \cosh \relax (x)^{5} + 3 \, \cosh \relax (x)^{3} + \cosh \relax (x)\right )} \sinh \relax (x) + 1\right )} \arctan \left (\cosh \relax (x) + \sinh \relax (x)\right ) + {\left (21 \, \cosh \relax (x)^{6} + 55 \, \cosh \relax (x)^{4} - 33 \, \cosh \relax (x)^{2} - 3\right )} \sinh \relax (x) - 3 \, \cosh \relax (x)}{4 \, {\left (\cosh \relax (x)^{8} + 8 \, \cosh \relax (x) \sinh \relax (x)^{7} + \sinh \relax (x)^{8} + 4 \, {\left (7 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{6} + 4 \, \cosh \relax (x)^{6} + 8 \, {\left (7 \, \cosh \relax (x)^{3} + 3 \, \cosh \relax (x)\right )} \sinh \relax (x)^{5} + 2 \, {\left (35 \, \cosh \relax (x)^{4} + 30 \, \cosh \relax (x)^{2} + 3\right )} \sinh \relax (x)^{4} + 6 \, \cosh \relax (x)^{4} + 8 \, {\left (7 \, \cosh \relax (x)^{5} + 10 \, \cosh \relax (x)^{3} + 3 \, \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 4 \, {\left (7 \, \cosh \relax (x)^{6} + 15 \, \cosh \relax (x)^{4} + 9 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{2} + 4 \, \cosh \relax (x)^{2} + 8 \, {\left (\cosh \relax (x)^{7} + 3 \, \cosh \relax (x)^{5} + 3 \, \cosh \relax (x)^{3} + \cosh \relax (x)\right )} \sinh \relax (x) + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.64, size = 60, normalized size = 2.31 \[ \frac {3}{16} \, \pi - \frac {3 \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} + 20 \, e^{\left (-x\right )} - 20 \, e^{x}}{4 \, {\left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4\right )}^{2}} + \frac {3}{8} \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.35, size = 21, normalized size = 0.81
method | result | size |
default | \(\left (\frac {\mathrm {sech}\relax (x )^{3}}{4}+\frac {3 \,\mathrm {sech}\relax (x )}{8}\right ) \tanh \relax (x )+\frac {3 \arctan \left ({\mathrm e}^{x}\right )}{4}\) | \(21\) |
risch | \(\frac {{\mathrm e}^{x} \left (3 \,{\mathrm e}^{6 x}+11 \,{\mathrm e}^{4 x}-11 \,{\mathrm e}^{2 x}-3\right )}{4 \left (1+{\mathrm e}^{2 x}\right )^{4}}+\frac {3 i \ln \left ({\mathrm e}^{x}+i\right )}{8}-\frac {3 i \ln \left ({\mathrm e}^{x}-i\right )}{8}\) | \(52\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.01, size = 61, normalized size = 2.35 \[ \frac {3 \, e^{\left (-x\right )} + 11 \, e^{\left (-3 \, x\right )} - 11 \, e^{\left (-5 \, x\right )} - 3 \, e^{\left (-7 \, x\right )}}{4 \, {\left (4 \, e^{\left (-2 \, x\right )} + 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} + e^{\left (-8 \, x\right )} + 1\right )}} - \frac {3}{4} \, \arctan \left (e^{\left (-x\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 22, normalized size = 0.85 \[ \frac {3\,\mathrm {atan}\left ({\mathrm {e}}^x\right )}{4}+\frac {3\,\mathrm {sinh}\relax (x)}{8\,{\mathrm {cosh}\relax (x)}^2}+\frac {\mathrm {sinh}\relax (x)}{4\,{\mathrm {cosh}\relax (x)}^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.83, size = 422, normalized size = 16.23 \[ \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} \]
Verification of antiderivative is not currently implemented for this CAS.
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