Optimal. Leaf size=19 \[ \frac{\cosh ^2(x)}{2 a}-\frac{\cosh (x)}{a} \]
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Rubi [A] time = 0.0432605, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2667} \[ \frac{\cosh ^2(x)}{2 a}-\frac{\cosh (x)}{a} \]
Antiderivative was successfully verified.
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Rule 2667
Rubi steps
\begin{align*} \int \frac{\sinh ^3(x)}{a+a \cosh (x)} \, dx &=-\frac{\operatorname{Subst}(\int (a-x) \, dx,x,a \cosh (x))}{a^3}\\ &=-\frac{\cosh (x)}{a}+\frac{\cosh ^2(x)}{2 a}\\ \end{align*}
Mathematica [A] time = 0.0116991, size = 13, normalized size = 0.68 \[ \frac{2 \sinh ^4\left (\frac{x}{2}\right )}{a} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.02, size = 47, normalized size = 2.5 \begin{align*} 8\,{\frac{1}{a} \left ( 1/16\, \left ( \tanh \left ( x/2 \right ) +1 \right ) ^{-2}-3/16\, \left ( \tanh \left ( x/2 \right ) +1 \right ) ^{-1}+1/16\, \left ( \tanh \left ( x/2 \right ) -1 \right ) ^{-2}+3/16\, \left ( \tanh \left ( x/2 \right ) -1 \right ) ^{-1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.11178, size = 49, normalized size = 2.58 \begin{align*} -\frac{{\left (4 \, e^{\left (-x\right )} - 1\right )} e^{\left (2 \, x\right )}}{8 \, a} - \frac{4 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )}}{8 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.88021, size = 58, normalized size = 3.05 \begin{align*} \frac{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 4 \, \cosh \left (x\right )}{4 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.106, size = 27, normalized size = 1.42 \begin{align*} \frac{2 \tanh ^{4}{\left (\frac{x}{2} \right )}}{a \tanh ^{4}{\left (\frac{x}{2} \right )} - 2 a \tanh ^{2}{\left (\frac{x}{2} \right )} + a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24729, size = 36, normalized size = 1.89 \begin{align*} -\frac{{\left (4 \, e^{x} - 1\right )} e^{\left (-2 \, x\right )} - e^{\left (2 \, x\right )} + 4 \, e^{x}}{8 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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