Optimal. Leaf size=19 \[ \frac{2 \cosh (x)}{3}-\frac{\sinh (x)}{3 (\coth (x)+1)} \]
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Rubi [A] time = 0.0355857, antiderivative size = 19, normalized size of antiderivative = 1., 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 3502
Rule 2638
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*}
Mathematica [A] time = 0.0476313, 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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Maple [B] time = 0.027, size = 40, normalized size = 2.1 \begin{align*} -{\frac{2}{3} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-3}}+ \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}+{\frac{1}{2} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}-{\frac{1}{2} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10248, size = 23, normalized size = 1.21 \begin{align*} \frac{1}{2} \, e^{\left (-x\right )} - \frac{1}{12} \, e^{\left (-3 \, x\right )} + \frac{1}{4} \, e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.47767, size = 99, normalized size = 5.21 \begin{align*} \frac{\cosh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 3}{6 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh{\left (x \right )}}{\coth{\left (x \right )} + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12542, size = 26, normalized size = 1.37 \begin{align*} \frac{1}{12} \,{\left (6 \, e^{\left (2 \, x\right )} - 1\right )} e^{\left (-3 \, x\right )} + \frac{1}{4} \, e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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