Optimal. Leaf size=21 \[ -\frac {1}{5} \text {sech}^5(x)+\frac {2 \text {sech}^3(x)}{3}-\text {sech}(x) \]
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Rubi [A] time = 0.02, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2606, 194} \[ -\frac {1}{5} \text {sech}^5(x)+\frac {2 \text {sech}^3(x)}{3}-\text {sech}(x) \]
Antiderivative was successfully verified.
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Rule 194
Rule 2606
Rubi steps
\begin {align*} \int \text {sech}(x) \tanh ^5(x) \, dx &=-\operatorname {Subst}\left (\int \left (-1+x^2\right )^2 \, dx,x,\text {sech}(x)\right )\\ &=-\operatorname {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\text {sech}(x)\right )\\ &=-\text {sech}(x)+\frac {2 \text {sech}^3(x)}{3}-\frac {\text {sech}^5(x)}{5}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 21, normalized size = 1.00 \[ -\frac {1}{5} \text {sech}^5(x)+\frac {2 \text {sech}^3(x)}{3}-\text {sech}(x) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 185, normalized size = 8.81 \[ -\frac {2 \, {\left (15 \, \cosh \relax (x)^{5} + 75 \, \cosh \relax (x) \sinh \relax (x)^{4} + 15 \, \sinh \relax (x)^{5} + 5 \, {\left (30 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{3} + 35 \, \cosh \relax (x)^{3} + 15 \, {\left (10 \, \cosh \relax (x)^{3} + 7 \, \cosh \relax (x)\right )} \sinh \relax (x)^{2} + {\left (75 \, \cosh \relax (x)^{4} + 15 \, \cosh \relax (x)^{2} + 38\right )} \sinh \relax (x) + 78 \, \cosh \relax (x)\right )}}{15 \, {\left (\cosh \relax (x)^{6} + 6 \, \cosh \relax (x) \sinh \relax (x)^{5} + \sinh \relax (x)^{6} + 3 \, {\left (5 \, \cosh \relax (x)^{2} + 2\right )} \sinh \relax (x)^{4} + 6 \, \cosh \relax (x)^{4} + 4 \, {\left (5 \, \cosh \relax (x)^{3} + 4 \, \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 3 \, {\left (5 \, \cosh \relax (x)^{4} + 12 \, \cosh \relax (x)^{2} + 5\right )} \sinh \relax (x)^{2} + 15 \, \cosh \relax (x)^{2} + 2 \, {\left (3 \, \cosh \relax (x)^{5} + 8 \, \cosh \relax (x)^{3} + 5 \, \cosh \relax (x)\right )} \sinh \relax (x) + 10\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 35, normalized size = 1.67 \[ -\frac {2 \, {\left (15 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{4} - 40 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 48\right )}}{15 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 28, normalized size = 1.33 \[ -\frac {\sinh ^{4}\relax (x )}{\cosh \relax (x )^{5}}-\frac {4 \left (\sinh ^{2}\relax (x )\right )}{3 \cosh \relax (x )^{5}}-\frac {8}{15 \cosh \relax (x )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 191, normalized size = 9.10 \[ -\frac {2 \, e^{\left (-x\right )}}{5 \, e^{\left (-2 \, x\right )} + 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} + 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} + 1} - \frac {8 \, e^{\left (-3 \, x\right )}}{3 \, {\left (5 \, e^{\left (-2 \, x\right )} + 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} + 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} + 1\right )}} - \frac {116 \, e^{\left (-5 \, x\right )}}{15 \, {\left (5 \, e^{\left (-2 \, x\right )} + 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} + 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} + 1\right )}} - \frac {8 \, e^{\left (-7 \, x\right )}}{3 \, {\left (5 \, e^{\left (-2 \, x\right )} + 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} + 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} + 1\right )}} - \frac {2 \, e^{\left (-9 \, x\right )}}{5 \, e^{\left (-2 \, x\right )} + 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} + 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.43, size = 129, normalized size = 6.14 \[ \frac {64\,{\mathrm {e}}^x}{5\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}-\frac {2\,{\mathrm {e}}^x}{{\mathrm {e}}^{2\,x}+1}-\frac {176\,{\mathrm {e}}^x}{15\,\left (3\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}+1\right )}-\frac {32\,{\mathrm {e}}^x}{5\,\left (5\,{\mathrm {e}}^{2\,x}+10\,{\mathrm {e}}^{4\,x}+10\,{\mathrm {e}}^{6\,x}+5\,{\mathrm {e}}^{8\,x}+{\mathrm {e}}^{10\,x}+1\right )}+\frac {16\,{\mathrm {e}}^x}{3\,\left (2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.47, size = 29, normalized size = 1.38 \[ - \frac {\tanh ^{4}{\relax (x )} \operatorname {sech}{\relax (x )}}{5} - \frac {4 \tanh ^{2}{\relax (x )} \operatorname {sech}{\relax (x )}}{15} - \frac {8 \operatorname {sech}{\relax (x )}}{15} \]
Verification of antiderivative is not currently implemented for this CAS.
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