Optimal. Leaf size=20 \[ \frac {\sinh (c+d x)}{d (\cosh (c+d x)+1)} \]
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Rubi [A] time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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*}
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Mathematica [A] time = 0.02, size = 14, normalized size = 0.70 \[ \frac {\tanh \left (\frac {1}{2} (c+d x)\right )}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 22, normalized size = 1.10 \[ -\frac {2}{d \cosh \left (d x + c\right ) + d \sinh \left (d x + c\right ) + d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 15, normalized size = 0.75 \[ -\frac {2}{d {\left (e^{\left (d x + c\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 14, normalized size = 0.70 \[ \frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 18, normalized size = 0.90 \[ \frac {2}{d {\left (e^{\left (-d x - c\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.89, size = 15, normalized size = 0.75 \[ -\frac {2}{d\,\left ({\mathrm {e}}^{c+d\,x}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.52, size = 17, normalized size = 0.85 \[ \begin {cases} \frac {\tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{d} & \text {for}\: d \neq 0 \\\frac {x}{\cosh {\relax (c )} + 1} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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