Optimal. Leaf size=16 \[ -\frac{\sinh ^3(x)}{3 (1-\cosh (x))^3} \]
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Rubi [A] time = 0.0328942, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2671} \[ -\frac{\sinh ^3(x)}{3 (1-\cosh (x))^3} \]
Antiderivative was successfully verified.
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Rule 2671
Rubi steps
\begin{align*} \int \frac{\sinh ^2(x)}{(1-\cosh (x))^3} \, dx &=-\frac{\sinh ^3(x)}{3 (1-\cosh (x))^3}\\ \end{align*}
Mathematica [A] time = 0.0285295, size = 12, normalized size = 0.75 \[ \frac{1}{3} \coth ^3\left (\frac{x}{2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 9, normalized size = 0.6 \begin{align*}{\frac{1}{3} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.10497, size = 66, normalized size = 4.12 \begin{align*} -\frac{2 \, e^{\left (-2 \, x\right )}}{3 \, e^{\left (-x\right )} - 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} - 1} - \frac{2}{3 \,{\left (3 \, e^{\left (-x\right )} - 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.05595, size = 126, normalized size = 7.88 \begin{align*} \frac{4 \,{\left (2 \, \cosh \left (x\right ) + \sinh \left (x\right )\right )}}{3 \,{\left (\cosh \left (x\right )^{2} + 2 \,{\left (\cosh \left (x\right ) - 1\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 4 \, \cosh \left (x\right ) + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.57552, size = 8, normalized size = 0.5 \begin{align*} \frac{1}{3 \tanh ^{3}{\left (\frac{x}{2} \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15602, size = 22, normalized size = 1.38 \begin{align*} \frac{2 \,{\left (3 \, e^{\left (2 \, x\right )} + 1\right )}}{3 \,{\left (e^{x} - 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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