Optimal. Leaf size=8 \[ \frac {1}{1-\cosh (x)} \]
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Rubi [A] time = 0.02, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2667, 32} \[ \frac {1}{1-\cosh (x)} \]
Antiderivative was successfully verified.
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Rule 32
Rule 2667
Rubi steps
\begin {align*} \int \frac {\sinh (x)}{(1-\cosh (x))^2} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{(1+x)^2} \, dx,x,-\cosh (x)\right )\\ &=\frac {1}{1-\cosh (x)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 12, normalized size = 1.50 \[ -\frac {1}{2} \text {csch}^2\left (\frac {x}{2}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.40, size = 31, normalized size = 3.88 \[ -\frac {2 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )}}{\cosh \relax (x)^{2} + 2 \, {\left (\cosh \relax (x) - 1\right )} \sinh \relax (x) + \sinh \relax (x)^{2} - 2 \, \cosh \relax (x) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 10, normalized size = 1.25 \[ -\frac {2 \, e^{x}}{{\left (e^{x} - 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 9, normalized size = 1.12 \[ \frac {1}{1-\cosh \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 8, normalized size = 1.00 \[ -\frac {1}{\cosh \relax (x) - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.91, size = 8, normalized size = 1.00 \[ -\frac {1}{\mathrm {cosh}\relax (x)-1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.54, size = 7, normalized size = 0.88 \[ - \frac {1}{\cosh {\relax (x )} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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