Integrand size = 4, antiderivative size = 26 \[ \int \text {sech}^5(x) \, dx=\frac {3}{8} \arctan (\sinh (x))+\frac {3}{8} \text {sech}(x) \tanh (x)+\frac {1}{4} \text {sech}^3(x) \tanh (x) \]
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Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3853, 3855} \[ \int \text {sech}^5(x) \, dx=\frac {3}{8} \arctan (\sinh (x))+\frac {1}{4} \tanh (x) \text {sech}^3(x)+\frac {3}{8} \tanh (x) \text {sech}(x) \]
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Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \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} \arctan (\sinh (x))+\frac {3}{8} \text {sech}(x) \tanh (x)+\frac {1}{4} \text {sech}^3(x) \tanh (x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \text {sech}^5(x) \, dx=\frac {3}{8} \arctan (\sinh (x))+\frac {3}{8} \text {sech}(x) \tanh (x)+\frac {1}{4} \text {sech}^3(x) \tanh (x) \]
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Time = 0.96 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.81
method | result | size |
default | \(\left (\frac {\operatorname {sech}\left (x \right )^{3}}{4}+\frac {3 \,\operatorname {sech}\left (x \right )}{8}\right ) \tanh \left (x \right )+\frac {3 \arctan \left ({\mathrm e}^{x}\right )}{4}\) | \(21\) |
parallelrisch | \(\frac {3 i \ln \left (i+\coth \left (x \right )-\operatorname {csch}\left (x \right )\right )}{8}-\frac {3 i \ln \left (-i+\coth \left (x \right )-\operatorname {csch}\left (x \right )\right )}{8}+\frac {3 \,\operatorname {sech}\left (x \right ) \tanh \left (x \right )}{8}+\frac {\operatorname {sech}\left (x \right )^{3} \tanh \left (x \right )}{4}\) | \(42\) |
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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Leaf count of result is larger than twice the leaf count of optimal. 461 vs. \(2 (20) = 40\).
Time = 0.24 (sec) , antiderivative size = 461, normalized size of antiderivative = 17.73 \[ \int \text {sech}^5(x) \, dx=\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 )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 422 vs. \(2 (27) = 54\).
Time = 1.17 (sec) , antiderivative size = 422, normalized size of antiderivative = 16.23 \[ \int \text {sech}^5(x) \, dx=\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} \]
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Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (20) = 40\).
Time = 0.28 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.35 \[ \int \text {sech}^5(x) \, dx=\frac {3 \, e^{\left (-x\right )} + 11 \, e^{\left (-3 \, x\right )} - 11 \, e^{\left (-5 \, x\right )} - 3 \, e^{\left (-7 \, x\right )}}{4 \, {\left (4 \, e^{\left (-2 \, x\right )} + 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} + e^{\left (-8 \, x\right )} + 1\right )}} - \frac {3}{4} \, \arctan \left (e^{\left (-x\right )}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (20) = 40\).
Time = 0.28 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.31 \[ \int \text {sech}^5(x) \, dx=\frac {3}{16} \, \pi - \frac {3 \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} + 20 \, e^{\left (-x\right )} - 20 \, e^{x}}{4 \, {\left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4\right )}^{2}} + \frac {3}{8} \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right ) \]
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Time = 0.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \text {sech}^5(x) \, dx=\frac {3\,\mathrm {atan}\left ({\mathrm {e}}^x\right )}{4}+\frac {3\,\mathrm {sinh}\left (x\right )}{8\,{\mathrm {cosh}\left (x\right )}^2}+\frac {\mathrm {sinh}\left (x\right )}{4\,{\mathrm {cosh}\left (x\right )}^4} \]
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