Optimal. Leaf size=13 \[ -\frac {8}{3 \left (1+e^{2 x}\right )^3} \]
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Rubi [A]
time = 0.01, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2320, 12, 267}
\begin {gather*} -\frac {8}{3 \left (e^{2 x}+1\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 267
Rule 2320
Rubi steps
\begin {align*} \int e^{-2 x} \text {sech}^4(x) \, dx &=\text {Subst}\left (\int \frac {16 x}{\left (1+x^2\right )^4} \, dx,x,e^x\right )\\ &=16 \text {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*}
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Mathematica [A]
time = 0.01, size = 13, normalized size = 1.00 \begin {gather*} -\frac {8}{3 \left (1+e^{2 x}\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(21\) vs.
\(2(10)=20\).
time = 0.10, size = 22, normalized size = 1.69
method | result | size |
risch | \(-\frac {8}{3 \left (1+{\mathrm e}^{2 x}\right )^{3}}\) | \(11\) |
default | \(2 \tanh \left (x \right )+\frac {1}{\cosh \left (x \right )^{2}}-\left (\frac {2}{3}+\frac {\mathrm {sech}\left (x \right )^{2}}{3}\right ) \tanh \left (x \right )\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 75 vs.
\(2 (10) = 20\).
time = 1.73, size = 75, normalized size = 5.77 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 102 vs.
\(2 (10) = 20\).
time = 0.66, size = 102, normalized size = 7.85 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{- 2 x}}{\cosh ^{4}{\left (x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.64, size = 10, normalized size = 0.77 \begin {gather*} -\frac {8}{3 \, {\left (e^{\left (2 \, x\right )} + 1\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.31, size = 19, normalized size = 1.46 \begin {gather*} -\frac {{\mathrm {e}}^{-3\,x}}{3\,{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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