Optimal. Leaf size=20 \[ -\frac {2}{1-\cosh (x)}-\log (1-\cosh (x)) \]
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Rubi [A]
time = 0.03, antiderivative size = 20, 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 = {2746, 45}
\begin {gather*} -\frac {2}{1-\cosh (x)}-\log (1-\cosh (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2746
Rubi steps
\begin {align*} \int \frac {\sinh ^3(x)}{(1-\cosh (x))^3} \, dx &=\text {Subst}\left (\int \frac {1-x}{(1+x)^2} \, dx,x,-\cosh (x)\right )\\ &=\text {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*}
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Mathematica [A]
time = 0.01, size = 27, normalized size = 1.35 \begin {gather*} \coth ^2\left (\frac {x}{2}\right )-2 \log \left (\cosh \left (\frac {x}{2}\right )\right )-2 \log \left (\tanh \left (\frac {x}{2}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.44, size = 17, normalized size = 0.85
method | result | size |
derivativedivides | \(-\ln \left (\cosh \left (x \right )-1\right )+\frac {2}{\cosh \left (x \right )-1}\) | \(17\) |
default | \(-\ln \left (\cosh \left (x \right )-1\right )+\frac {2}{\cosh \left (x \right )-1}\) | \(17\) |
risch | \(x +\frac {4 \,{\mathrm e}^{x}}{\left ({\mathrm e}^{x}-1\right )^{2}}-2 \ln \left ({\mathrm e}^{x}-1\right )\) | \(20\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 35, normalized size = 1.75 \begin {gather*} -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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 90 vs.
\(2 (18) = 36\).
time = 0.34, size = 90, normalized size = 4.50 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 126 vs.
\(2 (14) = 28\).
time = 0.28, size = 126, normalized size = 6.30 \begin {gather*} - \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 )}}{2 \cosh ^{2}{\left (x \right )} - 4 \cosh {\left (x \right )} + 2} + \frac {2 \cosh {\left (x \right )}}{2 \cosh ^{2}{\left (x \right )} - 4 \cosh {\left (x \right )} + 2} - \frac {2}{2 \cosh ^{2}{\left (x \right )} - 4 \cosh {\left (x \right )} + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 20, normalized size = 1.00 \begin {gather*} x + \frac {4 \, e^{x}}{{\left (e^{x} - 1\right )}^{2}} - 2 \, \log \left ({\left | e^{x} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.99, size = 16, normalized size = 0.80 \begin {gather*} \frac {2}{\mathrm {cosh}\left (x\right )-1}-\ln \left (\mathrm {cosh}\left (x\right )-1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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