Optimal. Leaf size=19 \[ \frac {\cosh ^2(x)}{2 a}-\frac {\cosh (x)}{a} \]
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Rubi [A] time = 0.04, antiderivative size = 19, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.01, size = 13, normalized size = 0.68 \[ \frac {2 \sinh ^4\left (\frac {x}{2}\right )}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 18, normalized size = 0.95 \[ \frac {\cosh \relax (x)^{2} + \sinh \relax (x)^{2} - 4 \, \cosh \relax (x)}{4 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 27, normalized size = 1.42 \[ -\frac {{\left (4 \, e^{x} - 1\right )} e^{\left (-2 \, x\right )} - e^{\left (2 \, x\right )} + 4 \, e^{x}}{8 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 47, normalized size = 2.47 \[ \frac {\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {3}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {3}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 36, normalized size = 1.89 \[ -\frac {{\left (4 \, e^{\left (-x\right )} - 1\right )} e^{\left (2 \, x\right )}}{8 \, a} - \frac {4 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )}}{8 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.94, size = 35, normalized size = 1.84 \[ \frac {{\mathrm {e}}^{-2\,x}}{8\,a}-\frac {{\mathrm {e}}^{-x}}{2\,a}+\frac {{\mathrm {e}}^{2\,x}}{8\,a}-\frac {{\mathrm {e}}^x}{2\,a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.73, size = 49, normalized size = 2.58 \[ \frac {4 \tanh ^{2}{\left (\frac {x}{2} \right )}}{a \tanh ^{4}{\left (\frac {x}{2} \right )} - 2 a \tanh ^{2}{\left (\frac {x}{2} \right )} + a} - \frac {2}{a \tanh ^{4}{\left (\frac {x}{2} \right )} - 2 a \tanh ^{2}{\left (\frac {x}{2} \right )} + a} \]
Verification of antiderivative is not currently implemented for this CAS.
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