Optimal. Leaf size=10 \[ -\frac {\text {sech}(x)}{1+\tanh (x)} \]
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Rubi [A]
time = 0.01, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {3569}
\begin {gather*} -\frac {\text {sech}(x)}{\tanh (x)+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 3569
Rubi steps
\begin {align*} \int \frac {\text {sech}(x)}{1+\tanh (x)} \, dx &=-\frac {\text {sech}(x)}{1+\tanh (x)}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 7, normalized size = 0.70 \begin {gather*} -\cosh (x)+\sinh (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.26, size = 11, normalized size = 1.10
method | result | size |
risch | \(-{\mathrm e}^{-x}\) | \(7\) |
gosper | \(-\frac {\mathrm {sech}\left (x \right )}{1+\tanh \left (x \right )}\) | \(11\) |
default | \(-\frac {2}{\tanh \left (\frac {x}{2}\right )+1}\) | \(11\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 6, normalized size = 0.60 \begin {gather*} -e^{\left (-x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 9, normalized size = 0.90 \begin {gather*} -\frac {1}{\cosh \left (x\right ) + \sinh \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.16, size = 8, normalized size = 0.80 \begin {gather*} - \frac {\operatorname {sech}{\left (x \right )}}{\tanh {\left (x \right )} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 6, normalized size = 0.60 \begin {gather*} -e^{\left (-x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 6, normalized size = 0.60 \begin {gather*} -{\mathrm {e}}^{-x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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