Optimal. Leaf size=13 \[ \frac{11 \text{sech}^3(x)}{3}-5 \text{sech}(x) \]
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Rubi [A] time = 0.035788, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {4339, 14} \[ \frac{11 \text{sech}^3(x)}{3}-5 \text{sech}(x) \]
Antiderivative was successfully verified.
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Rule 4339
Rule 14
Rubi steps
\begin{align*} \int \text{sech}(x) \left (5-11 \text{sech}^2(x)\right ) \tanh (x) \, dx &=\operatorname{Subst}\left (\int \frac{-11+5 x^2}{x^4} \, dx,x,\cosh (x)\right )\\ &=\operatorname{Subst}\left (\int \left (-\frac{11}{x^4}+\frac{5}{x^2}\right ) \, dx,x,\cosh (x)\right )\\ &=-5 \text{sech}(x)+\frac{11 \text{sech}^3(x)}{3}\\ \end{align*}
Mathematica [A] time = 0.007138, size = 13, normalized size = 1. \[ \frac{11 \text{sech}^3(x)}{3}-5 \text{sech}(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 12, normalized size = 0.9 \begin{align*} -5\,{\rm sech} \left (x\right )+{\frac{11\, \left ({\rm sech} \left (x\right ) \right ) ^{3}}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.03603, size = 31, normalized size = 2.38 \begin{align*} -\frac{10}{e^{\left (-x\right )} + e^{x}} + \frac{88}{3 \,{\left (e^{\left (-x\right )} + e^{x}\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.02602, size = 309, normalized size = 23.77 \begin{align*} -\frac{2 \,{\left (15 \, \cosh \left (x\right )^{3} + 45 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + 15 \, \sinh \left (x\right )^{3} +{\left (45 \, \cosh \left (x\right )^{2} - 29\right )} \sinh \left (x\right ) + \cosh \left (x\right )\right )}}{3 \,{\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \,{\left (3 \, \cosh \left (x\right )^{2} + 2\right )} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right )^{2} + 4 \,{\left (\cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.741429, size = 12, normalized size = 0.92 \begin{align*} \frac{11 \operatorname{sech}^{3}{\left (x \right )}}{3} - 5 \operatorname{sech}{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.11225, size = 32, normalized size = 2.46 \begin{align*} -\frac{2 \,{\left (15 \,{\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 44\right )}}{3 \,{\left (e^{\left (-x\right )} + e^{x}\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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