Optimal. Leaf size=24 \[ \frac {\text {sech}^3(x)}{3}+\frac {1}{2} \tan ^{-1}(\sinh (x))+\frac {1}{2} \tanh (x) \text {sech}(x) \]
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Rubi [A] time = 0.04, 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 = {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.
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Rule 3501
Rule 3768
Rule 3770
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 = 0.03, size = 24, normalized size = 1.00 \[ \frac {\text {sech}^3(x)}{3}+\tan ^{-1}\left (\tanh \left (\frac {x}{2}\right )\right )+\frac {1}{2} \tanh (x) \text {sech}(x) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 288, normalized size = 12.00 \[ \frac {3 \, \cosh \relax (x)^{5} + 15 \, \cosh \relax (x) \sinh \relax (x)^{4} + 3 \, \sinh \relax (x)^{5} + 2 \, {\left (15 \, \cosh \relax (x)^{2} + 4\right )} \sinh \relax (x)^{3} + 8 \, \cosh \relax (x)^{3} + 6 \, {\left (5 \, \cosh \relax (x)^{3} + 4 \, \cosh \relax (x)\right )} \sinh \relax (x)^{2} + 3 \, {\left (\cosh \relax (x)^{6} + 6 \, \cosh \relax (x) \sinh \relax (x)^{5} + \sinh \relax (x)^{6} + 3 \, {\left (5 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{4} + 3 \, \cosh \relax (x)^{4} + 4 \, {\left (5 \, \cosh \relax (x)^{3} + 3 \, \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 3 \, {\left (5 \, \cosh \relax (x)^{4} + 6 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{2} + 3 \, \cosh \relax (x)^{2} + 6 \, {\left (\cosh \relax (x)^{5} + 2 \, \cosh \relax (x)^{3} + \cosh \relax (x)\right )} \sinh \relax (x) + 1\right )} \arctan \left (\cosh \relax (x) + \sinh \relax (x)\right ) + 3 \, {\left (5 \, \cosh \relax (x)^{4} + 8 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x) - 3 \, \cosh \relax (x)}{3 \, {\left (\cosh \relax (x)^{6} + 6 \, \cosh \relax (x) \sinh \relax (x)^{5} + \sinh \relax (x)^{6} + 3 \, {\left (5 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{4} + 3 \, \cosh \relax (x)^{4} + 4 \, {\left (5 \, \cosh \relax (x)^{3} + 3 \, \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 3 \, {\left (5 \, \cosh \relax (x)^{4} + 6 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{2} + 3 \, \cosh \relax (x)^{2} + 6 \, {\left (\cosh \relax (x)^{5} + 2 \, \cosh \relax (x)^{3} + \cosh \relax (x)\right )} \sinh \relax (x) + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 31, normalized size = 1.29 \[ \frac {3 \, e^{\left (5 \, x\right )} + 8 \, e^{\left (3 \, x\right )} - 3 \, e^{x}}{3 \, {\left (e^{\left (2 \, x\right )} + 1\right )}^{3}} + \arctan \left (e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 41, normalized size = 1.71 \[ \frac {-\left (\tanh ^{5}\left (\frac {x}{2}\right )\right )+2 \left (\tanh ^{4}\left (\frac {x}{2}\right )\right )+\tanh \left (\frac {x}{2}\right )+\frac {2}{3}}{\left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{3}}+\arctan \left (\tanh \left (\frac {x}{2}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 49, normalized size = 2.04 \[ \frac {3 \, e^{\left (-x\right )} + 8 \, e^{\left (-3 \, x\right )} - 3 \, e^{\left (-5 \, x\right )}}{3 \, {\left (3 \, e^{\left (-2 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} + 1\right )}} - \arctan \left (e^{\left (-x\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 61, normalized size = 2.54 \[ \mathrm {atan}\left ({\mathrm {e}}^x\right )+\frac {{\mathrm {e}}^x}{{\mathrm {e}}^{2\,x}+1}-\frac {8\,{\mathrm {e}}^x}{3\,\left (3\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}+1\right )}+\frac {2\,{\mathrm {e}}^x}{3\,\left (2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {sech}^{5}{\relax (x )}}{\tanh {\relax (x )} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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