Optimal. Leaf size=20 \[ -\frac{2}{1-\cosh (x)}-\log (1-\cosh (x)) \]
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Rubi [A] time = 0.0404257, antiderivative size = 20, 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} \[ -\frac{2}{1-\cosh (x)}-\log (1-\cosh (x)) \]
Antiderivative was successfully verified.
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Rule 2667
Rule 43
Rubi steps
\begin{align*} \int \frac{\sinh ^3(x)}{(1-\cosh (x))^3} \, dx &=\operatorname{Subst}\left (\int \frac{1-x}{(1+x)^2} \, dx,x,-\cosh (x)\right )\\ &=\operatorname{Subst}\left (\int \left (\frac{1}{-1-x}+\frac{2}{(1+x)^2}\right ) \, dx,x,-\cosh (x)\right )\\ &=-\frac{2}{1-\cosh (x)}-\log (1-\cosh (x))\\ \end{align*}
Mathematica [A] time = 0.0124578, size = 27, normalized size = 1.35 \[ \coth ^2\left (\frac{x}{2}\right )-2 \log \left (\tanh \left (\frac{x}{2}\right )\right )-2 \log \left (\cosh \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 17, normalized size = 0.9 \begin{align*} 2\, \left ( -1+\cosh \left ( x \right ) \right ) ^{-1}-\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 = 1.15768, size = 47, normalized size = 2.35 \begin{align*} -x - \frac{4 \, e^{\left (-x\right )}}{2 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )} - 1} - 2 \, \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 = 1.88948, size = 335, normalized size = 16.75 \begin{align*} \frac{x \cosh \left (x\right )^{2} + x \sinh \left (x\right )^{2} - 2 \,{\left (x - 2\right )} \cosh \left (x\right ) - 2 \,{\left (\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\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) + 2 \,{\left (x \cosh \left (x\right ) - x + 2\right )} \sinh \left (x\right ) + x}{\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 [B] time = 0.949815, size = 194, normalized size = 9.7 \begin{align*} - \frac{2 \log{\left (\cosh{\left (x \right )} - 1 \right )} \cosh ^{2}{\left (x \right )}}{2 \cosh ^{2}{\left (x \right )} - 4 \cosh{\left (x \right )} + 2} + \frac{4 \log{\left (\cosh{\left (x \right )} - 1 \right )} \cosh{\left (x \right )}}{2 \cosh ^{2}{\left (x \right )} - 4 \cosh{\left (x \right )} + 2} - \frac{2 \log{\left (\cosh{\left (x \right )} - 1 \right )}}{2 \cosh ^{2}{\left (x \right )} - 4 \cosh{\left (x \right )} + 2} - \frac{\sinh ^{2}{\left (x \right )} \cosh ^{2}{\left (x \right )}}{2 \cosh ^{2}{\left (x \right )} - 4 \cosh{\left (x \right )} + 2} + \frac{2 \sinh ^{2}{\left (x \right )} \cosh{\left (x \right )}}{2 \cosh ^{2}{\left (x \right )} - 4 \cosh{\left (x \right )} + 2} + \frac{\cosh ^{4}{\left (x \right )}}{2 \cosh ^{2}{\left (x \right )} - 4 \cosh{\left (x \right )} + 2} - \frac{2 \cosh ^{3}{\left (x \right )}}{2 \cosh ^{2}{\left (x \right )} - 4 \cosh{\left (x \right )} + 2} + \frac{4 \cosh{\left (x \right )}}{2 \cosh ^{2}{\left (x \right )} - 4 \cosh{\left (x \right )} + 2} - \frac{3}{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 = 1.13683, size = 27, normalized size = 1.35 \begin{align*} x + \frac{4 \, e^{x}}{{\left (e^{x} - 1\right )}^{2}} - 2 \, \log \left ({\left | e^{x} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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