Optimal. Leaf size=10 \[ -\frac{\text{sech}(x)}{\tanh (x)+1} \]
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Rubi [A] time = 0.019921, antiderivative size = 10, normalized size of antiderivative = 1., 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 = {3488} \[ -\frac{\text{sech}(x)}{\tanh (x)+1} \]
Antiderivative was successfully verified.
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Rule 3488
Rubi steps
\begin{align*} \int \frac{\text{sech}(x)}{1+\tanh (x)} \, dx &=-\frac{\text{sech}(x)}{1+\tanh (x)}\\ \end{align*}
Mathematica [A] time = 0.0032273, size = 7, normalized size = 0.7 \[ \sinh (x)-\cosh (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 11, normalized size = 1.1 \begin{align*} -{\frac{{\rm sech} \left (x\right )}{1+\tanh \left ( x \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17869, size = 8, normalized size = 0.8 \begin{align*} -e^{\left (-x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.23128, size = 32, normalized size = 3.2 \begin{align*} -\frac{1}{\cosh \left (x\right ) + \sinh \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.381505, size = 8, normalized size = 0.8 \begin{align*} - \frac{\operatorname{sech}{\left (x \right )}}{\tanh{\left (x \right )} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19, size = 8, normalized size = 0.8 \begin{align*} -e^{\left (-x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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