Optimal. Leaf size=10 \[ -\frac {1}{2 (\cosh (x)+1)^2} \]
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Rubi [A] time = 0.02, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2667, 32} \[ -\frac {1}{2 (\cosh (x)+1)^2} \]
Antiderivative was successfully verified.
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Rule 32
Rule 2667
Rubi steps
\begin {align*} \int \frac {\sinh (x)}{(1+\cosh (x))^3} \, dx &=\operatorname {Subst}\left (\int \frac {1}{(1+x)^3} \, dx,x,\cosh (x)\right )\\ &=-\frac {1}{2 (1+\cosh (x))^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 12, normalized size = 1.20 \[ -\frac {1}{8} \text {sech}^4\left (\frac {x}{2}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.82, size = 55, normalized size = 5.50 \[ -\frac {2 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )}}{\cosh \relax (x)^{3} + {\left (3 \, \cosh \relax (x) + 4\right )} \sinh \relax (x)^{2} + \sinh \relax (x)^{3} + 4 \, \cosh \relax (x)^{2} + {\left (3 \, \cosh \relax (x)^{2} + 8 \, \cosh \relax (x) + 5\right )} \sinh \relax (x) + 7 \, \cosh \relax (x) + 4} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 12, normalized size = 1.20 \[ -\frac {2 \, e^{\left (2 \, x\right )}}{{\left (e^{x} + 1\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 9, normalized size = 0.90 \[ -\frac {1}{2 \left (1+\cosh \relax (x )\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.29, size = 8, normalized size = 0.80 \[ -\frac {1}{2 \, {\left (\cosh \relax (x) + 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.92, size = 8, normalized size = 0.80 \[ -\frac {1}{2\,{\left (\mathrm {cosh}\relax (x)+1\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.63, size = 15, normalized size = 1.50 \[ - \frac {1}{2 \cosh ^{2}{\relax (x )} + 4 \cosh {\relax (x )} + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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