Optimal. Leaf size=12 \[ \cosh (x)+2 \log (1-\cosh (x)) \]
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Rubi [A] time = 0.0372116, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2667, 43} \[ \cosh (x)+2 \log (1-\cosh (x)) \]
Antiderivative was successfully verified.
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Rule 2667
Rule 43
Rubi steps
\begin{align*} \int \frac{\sinh ^3(x)}{(1-\cosh (x))^2} \, dx &=\operatorname{Subst}\left (\int \frac{1-x}{1+x} \, dx,x,-\cosh (x)\right )\\ &=\operatorname{Subst}\left (\int \left (-1+\frac{2}{1+x}\right ) \, dx,x,-\cosh (x)\right )\\ &=\cosh (x)+2 \log (1-\cosh (x))\\ \end{align*}
Mathematica [A] time = 0.0169673, size = 13, normalized size = 1.08 \[ \cosh (x)+4 \log \left (\sinh \left (\frac{x}{2}\right )\right )-1 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 11, normalized size = 0.9 \begin{align*} \cosh \left ( x \right ) +2\,\ln \left ( -1+\cosh \left ( x \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02934, size = 31, normalized size = 2.58 \begin{align*} 2 \, x + \frac{1}{2} \, e^{\left (-x\right )} + \frac{1}{2} \, e^{x} + 4 \, \log \left (e^{\left (-x\right )} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.0958, size = 198, normalized size = 16.5 \begin{align*} -\frac{4 \, x \cosh \left (x\right ) - \cosh \left (x\right )^{2} - 8 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) + 2 \,{\left (2 \, x - \cosh \left (x\right )\right )} \sinh \left (x\right ) - \sinh \left (x\right )^{2} - 1}{2 \,{\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 [B] time = 0.643182, size = 70, normalized size = 5.83 \begin{align*} \frac{2 \log{\left (\cosh{\left (x \right )} - 1 \right )} \cosh{\left (x \right )}}{\cosh{\left (x \right )} - 1} - \frac{2 \log{\left (\cosh{\left (x \right )} - 1 \right )}}{\cosh{\left (x \right )} - 1} - \frac{\sinh ^{2}{\left (x \right )} \cosh{\left (x \right )}}{\cosh{\left (x \right )} - 1} + \frac{\cosh ^{3}{\left (x \right )}}{\cosh{\left (x \right )} - 1} + \frac{\cosh ^{2}{\left (x \right )}}{\cosh{\left (x \right )} - 1} - \frac{2}{\cosh{\left (x \right )} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12046, size = 30, normalized size = 2.5 \begin{align*} -2 \, x + \frac{1}{2} \, e^{\left (-x\right )} + \frac{1}{2} \, e^{x} + 4 \, \log \left ({\left | e^{x} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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