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 = 0.0186715, antiderivative size = 21, normalized size of antiderivative = 1., 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 2606
Rule 194
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*}
Mathematica [A] time = 0.012073, size = 21, normalized size = 1. \[ -\frac{1}{5} \text{sech}^5(x)+\frac{2 \text{sech}^3(x)}{3}-\text{sech}(x) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 46, normalized size = 2.2 \begin{align*} -{\frac{ \left ( \sinh \left ( x \right ) \right ) ^{4}}{ \left ( \cosh \left ( x \right ) \right ) ^{5}}}-{\frac{4\, \left ( \sinh \left ( x \right ) \right ) ^{2}}{5\, \left ( \cosh \left ( x \right ) \right ) ^{5}}}+{\frac{8\, \left ( \sinh \left ( x \right ) \right ) ^{2}}{15\, \left ( \cosh \left ( x \right ) \right ) ^{3}}}+{\frac{8\, \left ( \sinh \left ( x \right ) \right ) ^{2}}{15\,\cosh \left ( x \right ) }}-{\frac{8\,\cosh \left ( x \right ) }{15}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.28342, size = 258, normalized size = 12.29 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.83942, size = 620, normalized size = 29.52 \begin{align*} -\frac{2 \,{\left (15 \, \cosh \left (x\right )^{5} + 75 \, \cosh \left (x\right ) \sinh \left (x\right )^{4} + 15 \, \sinh \left (x\right )^{5} + 5 \,{\left (30 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{3} + 35 \, \cosh \left (x\right )^{3} + 15 \,{\left (10 \, \cosh \left (x\right )^{3} + 7 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} +{\left (75 \, \cosh \left (x\right )^{4} + 15 \, \cosh \left (x\right )^{2} + 38\right )} \sinh \left (x\right ) + 78 \, \cosh \left (x\right )\right )}}{15 \,{\left (\cosh \left (x\right )^{6} + 6 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6} + 3 \,{\left (5 \, \cosh \left (x\right )^{2} + 2\right )} \sinh \left (x\right )^{4} + 6 \, \cosh \left (x\right )^{4} + 4 \,{\left (5 \, \cosh \left (x\right )^{3} + 4 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \,{\left (5 \, \cosh \left (x\right )^{4} + 12 \, \cosh \left (x\right )^{2} + 5\right )} \sinh \left (x\right )^{2} + 15 \, \cosh \left (x\right )^{2} + 2 \,{\left (3 \, \cosh \left (x\right )^{5} + 8 \, \cosh \left (x\right )^{3} + 5 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 10\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.50233, size = 29, normalized size = 1.38 \begin{align*} - \frac{\tanh ^{4}{\left (x \right )} \operatorname{sech}{\left (x \right )}}{5} - \frac{4 \tanh ^{2}{\left (x \right )} \operatorname{sech}{\left (x \right )}}{15} - \frac{8 \operatorname{sech}{\left (x \right )}}{15} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.21732, size = 47, normalized size = 2.24 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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