Optimal. Leaf size=14 \[ \frac {\sinh ^3(x)}{3 (\cosh (x)+1)^3} \]
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Rubi [A] time = 0.03, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2671} \[ \frac {\sinh ^3(x)}{3 (\cosh (x)+1)^3} \]
Antiderivative was successfully verified.
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Rule 2671
Rubi steps
\begin {align*} \int \frac {\sinh ^2(x)}{(1+\cosh (x))^3} \, dx &=\frac {\sinh ^3(x)}{3 (1+\cosh (x))^3}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 12, normalized size = 0.86 \[ \frac {1}{3} \tanh ^3\left (\frac {x}{2}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.50, size = 33, normalized size = 2.36 \[ -\frac {4 \, {\left (2 \, \cosh \relax (x) + \sinh \relax (x)\right )}}{3 \, {\left (\cosh \relax (x)^{2} + 2 \, {\left (\cosh \relax (x) + 1\right )} \sinh \relax (x) + \sinh \relax (x)^{2} + 4 \, \cosh \relax (x) + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 16, normalized size = 1.14 \[ -\frac {2 \, {\left (3 \, e^{\left (2 \, x\right )} + 1\right )}}{3 \, {\left (e^{x} + 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 9, normalized size = 0.64 \[ \frac {\left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 49, normalized size = 3.50 \[ \frac {2 \, e^{\left (-2 \, x\right )}}{3 \, e^{\left (-x\right )} + 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} + 1} + \frac {2}{3 \, {\left (3 \, e^{\left (-x\right )} + 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.92, size = 16, normalized size = 1.14 \[ -\frac {2\,\left (3\,{\mathrm {e}}^{2\,x}+1\right )}{3\,{\left ({\mathrm {e}}^x+1\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.96, size = 7, normalized size = 0.50 \[ \frac {\tanh ^{3}{\left (\frac {x}{2} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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