Optimal. Leaf size=10 \[ -\frac{1}{2 (\cosh (x)+1)^2} \]
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Rubi [A] time = 0.0205692, antiderivative size = 10, normalized size of antiderivative = 1., 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 2667
Rule 32
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*}
Mathematica [A] time = 0.0088983, size = 12, normalized size = 1.2 \[ -\frac{1}{8} \text{sech}^4\left (\frac{x}{2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 9, normalized size = 0.9 \begin{align*} -{\frac{1}{2\, \left ( 1+\cosh \left ( x \right ) \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06343, size = 11, normalized size = 1.1 \begin{align*} -\frac{1}{2 \,{\left (\cosh \left (x\right ) + 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.04528, size = 197, normalized size = 19.7 \begin{align*} -\frac{2 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}}{\cosh \left (x\right )^{3} +{\left (3 \, \cosh \left (x\right ) + 4\right )} \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3} + 4 \, \cosh \left (x\right )^{2} +{\left (3 \, \cosh \left (x\right )^{2} + 8 \, \cosh \left (x\right ) + 5\right )} \sinh \left (x\right ) + 7 \, \cosh \left (x\right ) + 4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.81847, size = 15, normalized size = 1.5 \begin{align*} - \frac{1}{2 \cosh ^{2}{\left (x \right )} + 4 \cosh{\left (x \right )} + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18451, size = 16, normalized size = 1.6 \begin{align*} -\frac{2 \, e^{\left (2 \, x\right )}}{{\left (e^{x} + 1\right )}^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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