Optimal. Leaf size=8 \[ \frac{1}{1-\cosh (x)} \]
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Rubi [A] time = 0.0216567, antiderivative size = 8, 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} \[ \frac{1}{1-\cosh (x)} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 32
Rubi steps
\begin{align*} \int \frac{\sinh (x)}{(1-\cosh (x))^2} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{(1+x)^2} \, dx,x,-\cosh (x)\right )\\ &=\frac{1}{1-\cosh (x)}\\ \end{align*}
Mathematica [A] time = 0.0100208, size = 12, normalized size = 1.5 \[ -\frac{1}{2} \text{csch}^2\left (\frac{x}{2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 9, normalized size = 1.1 \begin{align*} \left ( 1-\cosh \left ( x \right ) \right ) ^{-1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03423, size = 11, normalized size = 1.38 \begin{align*} -\frac{1}{\cosh \left (x\right ) - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.06581, size = 122, normalized size = 15.25 \begin{align*} -\frac{2 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}}{\cosh \left (x\right )^{2} + 2 \,{\left (\cosh \left (x\right ) - 1\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.439663, size = 7, normalized size = 0.88 \begin{align*} - \frac{1}{\cosh{\left (x \right )} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1428, size = 14, normalized size = 1.75 \begin{align*} -\frac{2 \, e^{x}}{{\left (e^{x} - 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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