Optimal. Leaf size=20 \[ -\frac {2}{1-\cosh (x)}-\log (1-\cosh (x)) \]
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Rubi [A] time = 0.04, 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 = {2667, 43} \[ -\frac {2}{1-\cosh (x)}-\log (1-\cosh (x)) \]
Antiderivative was successfully verified.
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Rule 43
Rule 2667
Rubi steps
\begin {align*} \int \frac {\sinh ^3(x)}{(1-\cosh (x))^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*}
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Mathematica [A] time = 0.01, 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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fricas [B] time = 0.70, size = 90, normalized size = 4.50 \[ \frac {x \cosh \relax (x)^{2} + x \sinh \relax (x)^{2} - 2 \, {\left (x - 2\right )} \cosh \relax (x) - 2 \, {\left (\cosh \relax (x)^{2} + 2 \, {\left (\cosh \relax (x) - 1\right )} \sinh \relax (x) + \sinh \relax (x)^{2} - 2 \, \cosh \relax (x) + 1\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) - 1\right ) + 2 \, {\left (x \cosh \relax (x) - x + 2\right )} \sinh \relax (x) + x}{\cosh \relax (x)^{2} + 2 \, {\left (\cosh \relax (x) - 1\right )} \sinh \relax (x) + \sinh \relax (x)^{2} - 2 \, \cosh \relax (x) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 20, normalized size = 1.00 \[ x + \frac {4 \, e^{x}}{{\left (e^{x} - 1\right )}^{2}} - 2 \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 17, normalized size = 0.85 \[ -\ln \left (-1+\cosh \relax (x )\right )+\frac {2}{-1+\cosh \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 35, normalized size = 1.75 \[ -x - \frac {4 \, e^{\left (-x\right )}}{2 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )} - 1} - 2 \, \log \left (e^{\left (-x\right )} - 1\right ) \]
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 \[ \frac {2}{\mathrm {cosh}\relax (x)-1}-\ln \left (\mathrm {cosh}\relax (x)-1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.60, size = 126, normalized size = 6.30 \[ - \frac {2 \log {\left (\cosh {\relax (x )} - 1 \right )} \cosh ^{2}{\relax (x )}}{2 \cosh ^{2}{\relax (x )} - 4 \cosh {\relax (x )} + 2} + \frac {4 \log {\left (\cosh {\relax (x )} - 1 \right )} \cosh {\relax (x )}}{2 \cosh ^{2}{\relax (x )} - 4 \cosh {\relax (x )} + 2} - \frac {2 \log {\left (\cosh {\relax (x )} - 1 \right )}}{2 \cosh ^{2}{\relax (x )} - 4 \cosh {\relax (x )} + 2} + \frac {\sinh ^{2}{\relax (x )}}{2 \cosh ^{2}{\relax (x )} - 4 \cosh {\relax (x )} + 2} + \frac {2 \cosh {\relax (x )}}{2 \cosh ^{2}{\relax (x )} - 4 \cosh {\relax (x )} + 2} - \frac {2}{2 \cosh ^{2}{\relax (x )} - 4 \cosh {\relax (x )} + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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