Optimal. Leaf size=30 \[ -\frac {x}{4}+\frac {1}{8} \sinh (2 x)-\frac {1}{12} \sinh (3 x)+\frac {1}{20} \sinh (5 x) \]
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Rubi [A] time = 0.03, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {4355, 2637} \[ -\frac {x}{4}+\frac {1}{8} \sinh (2 x)-\frac {1}{12} \sinh (3 x)+\frac {1}{20} \sinh (5 x) \]
Antiderivative was successfully verified.
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Rule 2637
Rule 4355
Rubi steps
\begin {align*} \int \cosh \left (\frac {3 x}{2}\right ) \sinh (x) \sinh \left (\frac {5 x}{2}\right ) \, dx &=-\int \left (\frac {1}{4}-\frac {1}{4} \cosh (2 x)+\frac {1}{4} \cosh (3 x)-\frac {1}{4} \cosh (5 x)\right ) \, dx\\ &=-\frac {x}{4}+\frac {1}{4} \int \cosh (2 x) \, dx-\frac {1}{4} \int \cosh (3 x) \, dx+\frac {1}{4} \int \cosh (5 x) \, dx\\ &=-\frac {x}{4}+\frac {1}{8} \sinh (2 x)-\frac {1}{12} \sinh (3 x)+\frac {1}{20} \sinh (5 x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 30, normalized size = 1.00 \[ -\frac {x}{4}+\frac {1}{8} \sinh (2 x)-\frac {1}{12} \sinh (3 x)+\frac {1}{20} \sinh (5 x) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cosh \left (\frac {3 x}{2}\right ) \sinh (x) \sinh \left (\frac {5 x}{2}\right ) \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 1.23, size = 111, normalized size = 3.70 \[ 6 \, \cosh \left (\frac {1}{2} \, x\right )^{3} \sinh \left (\frac {1}{2} \, x\right )^{7} + \frac {1}{2} \, \cosh \left (\frac {1}{2} \, x\right ) \sinh \left (\frac {1}{2} \, x\right )^{9} + \frac {1}{10} \, {\left (126 \, \cosh \left (\frac {1}{2} \, x\right )^{5} - 5 \, \cosh \left (\frac {1}{2} \, x\right )\right )} \sinh \left (\frac {1}{2} \, x\right )^{5} + \frac {1}{6} \, {\left (36 \, \cosh \left (\frac {1}{2} \, x\right )^{7} - 10 \, \cosh \left (\frac {1}{2} \, x\right )^{3} + 3 \, \cosh \left (\frac {1}{2} \, x\right )\right )} \sinh \left (\frac {1}{2} \, x\right )^{3} + \frac {1}{2} \, {\left (\cosh \left (\frac {1}{2} \, x\right )^{9} - \cosh \left (\frac {1}{2} \, x\right )^{5} + \cosh \left (\frac {1}{2} \, x\right )^{3}\right )} \sinh \left (\frac {1}{2} \, x\right ) - \frac {1}{4} \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.64, size = 48, normalized size = 1.60 \[ \frac {1}{240} \, {\left (137 \, e^{\left (5 \, x\right )} - 15 \, e^{\left (3 \, x\right )} + 10 \, e^{\left (2 \, x\right )} - 6\right )} e^{\left (-5 \, x\right )} - \frac {1}{4} \, x + \frac {1}{40} \, e^{\left (5 \, x\right )} - \frac {1}{24} \, e^{\left (3 \, x\right )} + \frac {1}{16} \, e^{\left (2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 23, normalized size = 0.77
method | result | size |
default | \(-\frac {x}{4}+\frac {\sinh \left (2 x \right )}{8}-\frac {\sinh \left (3 x \right )}{12}+\frac {\sinh \left (5 x \right )}{20}\) | \(23\) |
risch | \(-\frac {x}{4}+\frac {{\mathrm e}^{5 x}}{40}-\frac {{\mathrm e}^{3 x}}{24}+\frac {{\mathrm e}^{2 x}}{16}-\frac {{\mathrm e}^{-2 x}}{16}+\frac {{\mathrm e}^{-3 x}}{24}-\frac {{\mathrm e}^{-5 x}}{40}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 42, normalized size = 1.40 \[ -\frac {1}{240} \, {\left (10 \, e^{\left (-2 \, x\right )} - 15 \, e^{\left (-3 \, x\right )} - 6\right )} e^{\left (5 \, x\right )} - \frac {1}{4} \, x - \frac {1}{16} \, e^{\left (-2 \, x\right )} + \frac {1}{24} \, e^{\left (-3 \, x\right )} - \frac {1}{40} \, e^{\left (-5 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.42, size = 40, normalized size = 1.33 \[ \frac {{\mathrm {e}}^{2\,x}}{16}-\frac {{\mathrm {e}}^{-2\,x}}{16}-\frac {x}{4}+\frac {{\mathrm {e}}^{-3\,x}}{24}-\frac {{\mathrm {e}}^{3\,x}}{24}-\frac {{\mathrm {e}}^{-5\,x}}{40}+\frac {{\mathrm {e}}^{5\,x}}{40} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 12.14, size = 138, normalized size = 4.60 \[ - \frac {x \sinh {\relax (x )} \sinh {\left (\frac {3 x}{2} \right )} \cosh {\left (\frac {5 x}{2} \right )}}{4} + \frac {x \sinh {\relax (x )} \sinh {\left (\frac {5 x}{2} \right )} \cosh {\left (\frac {3 x}{2} \right )}}{4} + \frac {x \sinh {\left (\frac {3 x}{2} \right )} \sinh {\left (\frac {5 x}{2} \right )} \cosh {\relax (x )}}{4} - \frac {x \cosh {\relax (x )} \cosh {\left (\frac {3 x}{2} \right )} \cosh {\left (\frac {5 x}{2} \right )}}{4} - \frac {\sinh {\relax (x )} \sinh {\left (\frac {3 x}{2} \right )} \sinh {\left (\frac {5 x}{2} \right )}}{12} + \frac {7 \sinh {\relax (x )} \cosh {\left (\frac {3 x}{2} \right )} \cosh {\left (\frac {5 x}{2} \right )}}{20} - \frac {\sinh {\left (\frac {3 x}{2} \right )} \cosh {\relax (x )} \cosh {\left (\frac {5 x}{2} \right )}}{15} \]
Verification of antiderivative is not currently implemented for this CAS.
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