Optimal. Leaf size=12 \[ \frac{1}{2 (1-\cosh (x))^2} \]
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Rubi [A] time = 0.0211197, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2667, 32} \[ \frac{1}{2 (1-\cosh (x))^2} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 32
Rubi steps
\begin{align*} \int \frac{\sinh (x)}{(1-\cosh (x))^3} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{(1+x)^3} \, dx,x,-\cosh (x)\right )\\ &=\frac{1}{2 (1-\cosh (x))^2}\\ \end{align*}
Mathematica [A] time = 0.0103202, size = 12, normalized size = 1. \[ \frac{1}{8} \text{csch}^4\left (\frac{x}{2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 11, normalized size = 0.9 \begin{align*}{\frac{1}{2\, \left ( 1-\cosh \left ( x \right ) \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06197, size = 11, normalized size = 0.92 \begin{align*} \frac{1}{2 \,{\left (\cosh \left (x\right ) - 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.13166, size = 196, normalized size = 16.33 \begin{align*} \frac{2 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}}{\cosh \left (x\right )^{3} +{\left (3 \, \cosh \left (x\right ) - 4\right )} \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3} - 4 \, \cosh \left (x\right )^{2} +{\left (3 \, \cosh \left (x\right )^{2} - 8 \, \cosh \left (x\right ) + 5\right )} \sinh \left (x\right ) + 7 \, \cosh \left (x\right ) - 4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.842427, size = 14, normalized size = 1.17 \begin{align*} \frac{1}{2 \cosh ^{2}{\left (x \right )} - 4 \cosh{\left (x \right )} + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14395, size = 16, normalized size = 1.33 \begin{align*} \frac{2 \, e^{\left (2 \, x\right )}}{{\left (e^{x} - 1\right )}^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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