Optimal. Leaf size=20 \[ \frac{\sinh (c+d x)}{d (\cosh (c+d x)+1)} \]
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Rubi [A] time = 0.0100075, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2648} \[ \frac{\sinh (c+d x)}{d (\cosh (c+d x)+1)} \]
Antiderivative was successfully verified.
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Rule 2648
Rubi steps
\begin{align*} \int \frac{1}{1+\cosh (c+d x)} \, dx &=\frac{\sinh (c+d x)}{d (1+\cosh (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.0141596, size = 14, normalized size = 0.7 \[ \frac{\tanh \left (\frac{1}{2} (c+d x)\right )}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 14, normalized size = 0.7 \begin{align*}{\frac{1}{d}\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16488, size = 24, normalized size = 1.2 \begin{align*} \frac{2}{d{\left (e^{\left (-d x - c\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73141, size = 59, normalized size = 2.95 \begin{align*} -\frac{2}{d \cosh \left (d x + c\right ) + d \sinh \left (d x + c\right ) + d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.592076, size = 17, normalized size = 0.85 \begin{align*} \begin{cases} \frac{\tanh{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{d} & \text{for}\: d \neq 0 \\\frac{x}{\cosh{\left (c \right )} + 1} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16337, size = 20, normalized size = 1. \begin{align*} -\frac{2}{d{\left (e^{\left (d x + c\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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