Optimal. Leaf size=19 \[ \frac {2 \cosh (x)}{3}-\frac {\sinh (x)}{3 (\coth (x)+1)} \]
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Rubi [A] time = 0.04, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3502, 2638} \[ \frac {2 \cosh (x)}{3}-\frac {\sinh (x)}{3 (\coth (x)+1)} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3502
Rubi steps
\begin {align*} \int \frac {\sinh (x)}{1+\coth (x)} \, dx &=-\frac {\sinh (x)}{3 (1+\coth (x))}+\frac {2}{3} \int \sinh (x) \, dx\\ &=\frac {2 \cosh (x)}{3}-\frac {\sinh (x)}{3 (1+\coth (x))}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 21, normalized size = 1.11 \[ \frac {1}{12} \left (4 \sinh ^3(x)+9 \cosh (x)-\cosh (3 x)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 25, normalized size = 1.32 \[ \frac {\cosh \relax (x)^{2} + 4 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + 3}{6 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 19, normalized size = 1.00 \[ \frac {1}{12} \, {\left (6 \, e^{\left (2 \, x\right )} - 1\right )} e^{\left (-3 \, x\right )} + \frac {1}{4} \, e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 40, normalized size = 2.11 \[ -\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {2}{3 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {1}{2 \tanh \left (\frac {x}{2}\right )+2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 17, normalized size = 0.89 \[ \frac {1}{2} \, e^{\left (-x\right )} - \frac {1}{12} \, e^{\left (-3 \, x\right )} + \frac {1}{4} \, e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.24, size = 17, normalized size = 0.89 \[ \frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^{-3\,x}}{12}+\frac {{\mathrm {e}}^x}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sinh {\relax (x )}}{\coth {\relax (x )} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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