Optimal. Leaf size=13 \[ -\frac{8}{3 \left (e^{2 x}+1\right )^3} \]
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Rubi [A] time = 0.01701, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {2282, 12, 261} \[ -\frac{8}{3 \left (e^{2 x}+1\right )^3} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 12
Rule 261
Rubi steps
\begin{align*} \int e^{-2 x} \text{sech}^4(x) \, dx &=\operatorname{Subst}\left (\int \frac{16 x}{\left (1+x^2\right )^4} \, dx,x,e^x\right )\\ &=16 \operatorname{Subst}\left (\int \frac{x}{\left (1+x^2\right )^4} \, dx,x,e^x\right )\\ &=-\frac{8}{3 \left (1+e^{2 x}\right )^3}\\ \end{align*}
Mathematica [A] time = 0.0095893, size = 13, normalized size = 1. \[ -\frac{8}{3 \left (e^{2 x}+1\right )^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.043, size = 52, normalized size = 4. \begin{align*} -2\,{\frac{- \left ( \tanh \left ( x/2 \right ) \right ) ^{5}+2\, \left ( \tanh \left ( x/2 \right ) \right ) ^{4}-10/3\, \left ( \tanh \left ( x/2 \right ) \right ) ^{3}+2\, \left ( \tanh \left ( x/2 \right ) \right ) ^{2}-\tanh \left ( x/2 \right ) }{ \left ( \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.976321, size = 101, normalized size = 7.77 \begin{align*} \frac{8 \, e^{\left (-2 \, x\right )}}{3 \, e^{\left (-2 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} + 1} + \frac{8 \, e^{\left (-4 \, x\right )}}{3 \, e^{\left (-2 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} + 1} + \frac{8}{3 \,{\left (3 \, e^{\left (-2 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.11093, size = 338, normalized size = 26. \begin{align*} -\frac{8}{3 \,{\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} + 1\right )} \sinh \left (x\right )^{4} + 3 \, \cosh \left (x\right )^{4} + 4 \,{\left (5 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \,{\left (5 \, \cosh \left (x\right )^{4} + 6 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 3 \, \cosh \left (x\right )^{2} + 6 \,{\left (\cosh \left (x\right )^{5} + 2 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{- 2 x}}{\cosh ^{4}{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08337, size = 14, normalized size = 1.08 \begin{align*} -\frac{8}{3 \,{\left (e^{\left (2 \, x\right )} + 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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