Optimal. Leaf size=18 \[ \frac {x}{2}-\frac {\sinh (x)}{2}+\frac {\cosh (x)}{2} \]
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Rubi [A] time = 0.03, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3131} \[ \frac {x}{2}-\frac {\sinh (x)}{2}+\frac {\cosh (x)}{2} \]
Antiderivative was successfully verified.
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Rule 3131
Rubi steps
\begin {align*} \int \frac {\sinh (x)}{1+\cosh (x)+\sinh (x)} \, dx &=\frac {x}{2}+\frac {\cosh (x)}{2}-\frac {\sinh (x)}{2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 18, normalized size = 1.00 \[ \frac {x}{2}-\frac {\sinh (x)}{2}+\frac {\cosh (x)}{2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 19, normalized size = 1.06 \[ \frac {x \cosh \relax (x) + x \sinh \relax (x) + 1}{2 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 10, normalized size = 0.56 \[ \frac {1}{2} \, x + \frac {1}{2} \, e^{\left (-x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.17, size = 28, normalized size = 1.56 \[ -\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2}+\frac {1}{\tanh \left (\frac {x}{2}\right )+1}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.74, size = 10, normalized size = 0.56 \[ \frac {1}{2} \, x + \frac {1}{2} \, e^{\left (-x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 10, normalized size = 0.56 \[ \frac {x}{2}+\frac {{\mathrm {e}}^{-x}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.56, size = 34, normalized size = 1.89 \[ \frac {x \tanh {\left (\frac {x}{2} \right )}}{2 \tanh {\left (\frac {x}{2} \right )} + 2} + \frac {x}{2 \tanh {\left (\frac {x}{2} \right )} + 2} + \frac {2}{2 \tanh {\left (\frac {x}{2} \right )} + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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