Optimal. Leaf size=17 \[ \frac {1}{4} \cosh (2 x)+\frac {1}{8} \cosh (4 x) \]
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Rubi [A]
time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4369}
\begin {gather*} \frac {1}{4} \cosh (2 x)+\frac {1}{8} \cosh (4 x) \end {gather*}
Antiderivative was successfully verified.
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Rule 4369
Rubi steps
\begin {align*} \int \cosh (x) \sinh (3 x) \, dx &=\frac {1}{4} \cosh (2 x)+\frac {1}{8} \cosh (4 x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 17, normalized size = 1.00 \begin {gather*} \frac {\cosh ^2(x)}{2}+\frac {1}{8} \cosh (4 x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.06, size = 13, normalized size = 0.76
method | result | size |
derivativedivides | \(\frac {\left (4 \left (\cosh ^{2}\left (x \right )\right )-1\right )^{2}}{16}\) | \(13\) |
default | \(\frac {\left (4 \left (\cosh ^{2}\left (x \right )\right )-1\right )^{2}}{16}\) | \(13\) |
risch | \(\frac {{\mathrm e}^{4 x}}{16}+\frac {{\mathrm e}^{2 x}}{8}+\frac {{\mathrm e}^{-2 x}}{8}+\frac {{\mathrm e}^{-4 x}}{16}\) | \(26\) |
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. 27 vs.
\(2 (13) = 26\).
time = 0.26, size = 27, normalized size = 1.59 \begin {gather*} \frac {1}{16} \, {\left (2 \, e^{\left (-2 \, x\right )} + 1\right )} e^{\left (4 \, x\right )} + \frac {1}{8} \, e^{\left (-2 \, x\right )} + \frac {1}{16} \, e^{\left (-4 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 33 vs.
\(2 (13) = 26\).
time = 0.36, size = 33, normalized size = 1.94 \begin {gather*} \frac {1}{8} \, \cosh \left (x\right )^{4} + \frac {1}{8} \, \sinh \left (x\right )^{4} + \frac {1}{4} \, {\left (3 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + \frac {1}{4} \, \cosh \left (x\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.12, size = 20, normalized size = 1.18 \begin {gather*} - \frac {\sinh {\left (x \right )} \sinh {\left (3 x \right )}}{8} + \frac {3 \cosh {\left (x \right )} \cosh {\left (3 x \right )}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 26, normalized size = 1.53 \begin {gather*} \frac {1}{16} \, {\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}^{2} + \frac {1}{8} \, e^{\left (2 \, x\right )} + \frac {1}{8} \, e^{\left (-2 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.45, size = 11, normalized size = 0.65 \begin {gather*} {\mathrm {cosh}\left (x\right )}^4-\frac {{\mathrm {cosh}\left (x\right )}^2}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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