Optimal. Leaf size=12 \[ \cosh (x)+2 \log (1-\cosh (x)) \]
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Rubi [A] time = 0.04, antiderivative size = 12, normalized size of antiderivative = 1.00, 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 43
Rule 2667
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*}
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Mathematica [A] time = 0.02, 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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fricas [B] time = 1.09, size = 54, normalized size = 4.50 \[ -\frac {4 \, x \cosh \relax (x) - \cosh \relax (x)^{2} - 8 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) - 1\right ) + 2 \, {\left (2 \, x - \cosh \relax (x)\right )} \sinh \relax (x) - \sinh \relax (x)^{2} - 1}{2 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 22, normalized size = 1.83 \[ -2 \, x + \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} + 4 \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 11, normalized size = 0.92 \[ \cosh \relax (x )+2 \ln \left (-1+\cosh \relax (x )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 23, normalized size = 1.92 \[ 2 \, x + \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} + 4 \, \log \left (e^{\left (-x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.94, size = 10, normalized size = 0.83 \[ 2\,\ln \left (\mathrm {cosh}\relax (x)-1\right )+\mathrm {cosh}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.53, size = 58, normalized size = 4.83 \[ \frac {2 \log {\left (\cosh {\relax (x )} - 1 \right )} \cosh {\relax (x )}}{\cosh {\relax (x )} - 1} - \frac {2 \log {\left (\cosh {\relax (x )} - 1 \right )}}{\cosh {\relax (x )} - 1} - \frac {\sinh ^{2}{\relax (x )}}{\cosh {\relax (x )} - 1} + \frac {2 \cosh ^{2}{\relax (x )}}{\cosh {\relax (x )} - 1} - \frac {2}{\cosh {\relax (x )} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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