Optimal. Leaf size=17 \[ \frac{\tanh ^6(x)}{6}-\frac{\tanh ^8(x)}{8} \]
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Rubi [A] time = 0.0781993, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4120, 2607, 14} \[ \frac{\tanh ^6(x)}{6}-\frac{\tanh ^8(x)}{8} \]
Antiderivative was successfully verified.
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Rule 4120
Rule 2607
Rule 14
Rubi steps
\begin{align*} \int \text{sech}^4(x) \left (-1+\text{sech}^2(x)\right )^2 \tanh (x) \, dx &=\int \text{sech}^4(x) \tanh ^5(x) \, dx\\ &=-\operatorname{Subst}\left (\int x^5 \left (1+x^2\right ) \, dx,x,i \tanh (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (x^5+x^7\right ) \, dx,x,i \tanh (x)\right )\\ &=\frac{\tanh ^6(x)}{6}-\frac{\tanh ^8(x)}{8}\\ \end{align*}
Mathematica [A] time = 0.0152839, size = 25, normalized size = 1.47 \[ -\frac{1}{8} \text{sech}^8(x)+\frac{\text{sech}^6(x)}{3}-\frac{\text{sech}^4(x)}{4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 20, normalized size = 1.2 \begin{align*} -{\frac{ \left ({\rm sech} \left (x\right ) \right ) ^{8}}{8}}+{\frac{ \left ({\rm sech} \left (x\right ) \right ) ^{6}}{3}}-{\frac{ \left ({\rm sech} \left (x\right ) \right ) ^{4}}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.02699, size = 46, normalized size = 2.71 \begin{align*} -\frac{4}{{\left (e^{\left (-x\right )} + e^{x}\right )}^{4}} + \frac{64}{3 \,{\left (e^{\left (-x\right )} + e^{x}\right )}^{6}} - \frac{32}{{\left (e^{\left (-x\right )} + e^{x}\right )}^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.99374, size = 1157, normalized size = 68.06 \begin{align*} -\frac{4 \,{\left (3 \, \cosh \left (x\right )^{6} + 18 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + 3 \, \sinh \left (x\right )^{6} +{\left (45 \, \cosh \left (x\right )^{2} - 4\right )} \sinh \left (x\right )^{4} - 4 \, \cosh \left (x\right )^{4} + 4 \,{\left (15 \, \cosh \left (x\right )^{3} - 4 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} +{\left (45 \, \cosh \left (x\right )^{4} - 24 \, \cosh \left (x\right )^{2} + 13\right )} \sinh \left (x\right )^{2} + 13 \, \cosh \left (x\right )^{2} + 2 \,{\left (9 \, \cosh \left (x\right )^{5} - 8 \, \cosh \left (x\right )^{3} + 7 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) - 4\right )}}{3 \,{\left (\cosh \left (x\right )^{10} + 10 \, \cosh \left (x\right ) \sinh \left (x\right )^{9} + \sinh \left (x\right )^{10} +{\left (45 \, \cosh \left (x\right )^{2} + 8\right )} \sinh \left (x\right )^{8} + 8 \, \cosh \left (x\right )^{8} + 8 \,{\left (15 \, \cosh \left (x\right )^{3} + 8 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{7} +{\left (210 \, \cosh \left (x\right )^{4} + 224 \, \cosh \left (x\right )^{2} + 29\right )} \sinh \left (x\right )^{6} + 29 \, \cosh \left (x\right )^{6} + 2 \,{\left (126 \, \cosh \left (x\right )^{5} + 224 \, \cosh \left (x\right )^{3} + 81 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} +{\left (210 \, \cosh \left (x\right )^{6} + 560 \, \cosh \left (x\right )^{4} + 435 \, \cosh \left (x\right )^{2} + 64\right )} \sinh \left (x\right )^{4} + 64 \, \cosh \left (x\right )^{4} + 4 \,{\left (30 \, \cosh \left (x\right )^{7} + 112 \, \cosh \left (x\right )^{5} + 135 \, \cosh \left (x\right )^{3} + 48 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} +{\left (45 \, \cosh \left (x\right )^{8} + 224 \, \cosh \left (x\right )^{6} + 435 \, \cosh \left (x\right )^{4} + 384 \, \cosh \left (x\right )^{2} + 98\right )} \sinh \left (x\right )^{2} + 98 \, \cosh \left (x\right )^{2} + 2 \,{\left (5 \, \cosh \left (x\right )^{9} + 32 \, \cosh \left (x\right )^{7} + 81 \, \cosh \left (x\right )^{5} + 96 \, \cosh \left (x\right )^{3} + 42 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 56\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 18.3691, size = 19, normalized size = 1.12 \begin{align*} - \frac{\operatorname{sech}^{8}{\left (x \right )}}{8} + \frac{\operatorname{sech}^{6}{\left (x \right )}}{3} - \frac{\operatorname{sech}^{4}{\left (x \right )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13431, size = 55, normalized size = 3.24 \begin{align*} -\frac{4 \,{\left (3 \, e^{\left (12 \, x\right )} - 4 \, e^{\left (10 \, x\right )} + 10 \, e^{\left (8 \, x\right )} - 4 \, e^{\left (6 \, x\right )} + 3 \, e^{\left (4 \, x\right )}\right )}}{3 \,{\left (e^{\left (2 \, x\right )} + 1\right )}^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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