Optimal. Leaf size=7 \[ -\frac {\cosh (x)}{a} \]
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Rubi [A]
time = 0.03, antiderivative size = 7, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3254, 2718}
\begin {gather*} -\frac {\cosh (x)}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 2718
Rule 3254
Rubi steps
\begin {align*} \int \frac {\sinh ^3(x)}{a-a \cosh ^2(x)} \, dx &=-\frac {\int \sinh (x) \, dx}{a}\\ &=-\frac {\cosh (x)}{a}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 7, normalized size = 1.00 \begin {gather*} -\frac {\cosh (x)}{a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.41, size = 8, normalized size = 1.14
method | result | size |
derivativedivides | \(-\frac {\cosh \left (x \right )}{a}\) | \(8\) |
default | \(-\frac {\cosh \left (x \right )}{a}\) | \(8\) |
risch | \(-\frac {{\mathrm e}^{x}}{2 a}-\frac {{\mathrm e}^{-x}}{2 a}\) | \(18\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 17 vs.
\(2 (7) = 14\).
time = 0.26, size = 17, normalized size = 2.43 \begin {gather*} -\frac {e^{\left (-x\right )}}{2 \, a} - \frac {e^{x}}{2 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 7, normalized size = 1.00 \begin {gather*} -\frac {\cosh \left (x\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.53, size = 10, normalized size = 1.43 \begin {gather*} \frac {2}{a \tanh ^{2}{\left (\frac {x}{2} \right )} - a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 12, normalized size = 1.71 \begin {gather*} -\frac {e^{\left (-x\right )} + e^{x}}{2 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.91, size = 7, normalized size = 1.00 \begin {gather*} -\frac {\mathrm {cosh}\left (x\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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