Integrand size = 15, antiderivative size = 30 \[ \int \cosh \left (\frac {3 x}{2}\right ) \sinh (x) \sinh \left (\frac {5 x}{2}\right ) \, dx=-\frac {x}{4}+\frac {1}{8} \sinh (2 x)-\frac {1}{12} \sinh (3 x)+\frac {1}{20} \sinh (5 x) \]
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Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {4440, 2717} \[ \int \cosh \left (\frac {3 x}{2}\right ) \sinh (x) \sinh \left (\frac {5 x}{2}\right ) \, dx=-\frac {x}{4}+\frac {1}{8} \sinh (2 x)-\frac {1}{12} \sinh (3 x)+\frac {1}{20} \sinh (5 x) \]
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Rule 2717
Rule 4440
Rubi steps \begin{align*} \text {integral}& = -\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*}
Time = 0.04 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \cosh \left (\frac {3 x}{2}\right ) \sinh (x) \sinh \left (\frac {5 x}{2}\right ) \, dx=-\frac {x}{4}+\frac {1}{8} \sinh (2 x)-\frac {1}{12} \sinh (3 x)+\frac {1}{20} \sinh (5 x) \]
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Time = 2.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 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 | \(-\frac {x}{4}+\frac {{\mathrm e}^{5 x}}{40}-\frac {{\mathrm e}^{3 x}}{24}+\frac {{\mathrm e}^{2 x}}{16}-\frac {{\mathrm e}^{-2 x}}{16}+\frac {{\mathrm e}^{-3 x}}{24}-\frac {{\mathrm e}^{-5 x}}{40}\) | \(41\) |
parallelrisch | \(\frac {\left (-40 \cosh \left (\frac {x}{2}\right )+40\right ) \ln \left (1-\tanh \left (\frac {3 x}{4}\right )\right )+\left (40 \cosh \left (\frac {x}{2}\right )-40\right ) \ln \left (\tanh \left (\frac {3 x}{4}\right )+1\right )-120 x \cosh \left (\frac {x}{2}\right )+120 x +6 \sinh \left (\frac {11 x}{2}\right )-30 \sinh \left (2 x \right )+20 \sinh \left (3 x \right )-12 \sinh \left (5 x \right )+15 \sinh \left (\frac {3 x}{2}\right )+5 \sinh \left (\frac {5 x}{2}\right )-10 \sinh \left (\frac {7 x}{2}\right )+6 \sinh \left (\frac {9 x}{2}\right )}{-240+240 \cosh \left (\frac {x}{2}\right )}\) | \(105\) |
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Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (22) = 44\).
Time = 0.24 (sec) , antiderivative size = 111, normalized size of antiderivative = 3.70 \[ \int \cosh \left (\frac {3 x}{2}\right ) \sinh (x) \sinh \left (\frac {5 x}{2}\right ) \, dx=6 \, \cosh \left (\frac {1}{2} \, x\right )^{3} \sinh \left (\frac {1}{2} \, x\right )^{7} + \frac {1}{2} \, \cosh \left (\frac {1}{2} \, x\right ) \sinh \left (\frac {1}{2} \, x\right )^{9} + \frac {1}{10} \, {\left (126 \, \cosh \left (\frac {1}{2} \, x\right )^{5} - 5 \, \cosh \left (\frac {1}{2} \, x\right )\right )} \sinh \left (\frac {1}{2} \, x\right )^{5} + \frac {1}{6} \, {\left (36 \, \cosh \left (\frac {1}{2} \, x\right )^{7} - 10 \, \cosh \left (\frac {1}{2} \, x\right )^{3} + 3 \, \cosh \left (\frac {1}{2} \, x\right )\right )} \sinh \left (\frac {1}{2} \, x\right )^{3} + \frac {1}{2} \, {\left (\cosh \left (\frac {1}{2} \, x\right )^{9} - \cosh \left (\frac {1}{2} \, x\right )^{5} + \cosh \left (\frac {1}{2} \, x\right )^{3}\right )} \sinh \left (\frac {1}{2} \, x\right ) - \frac {1}{4} \, x \]
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Leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (22) = 44\).
Time = 0.94 (sec) , antiderivative size = 139, normalized size of antiderivative = 4.63 \[ \int \cosh \left (\frac {3 x}{2}\right ) \sinh (x) \sinh \left (\frac {5 x}{2}\right ) \, dx=- \frac {x \sinh {\left (x \right )} \sinh {\left (\frac {3 x}{2} \right )} \cosh {\left (\frac {5 x}{2} \right )}}{4} + \frac {x \sinh {\left (x \right )} \sinh {\left (\frac {5 x}{2} \right )} \cosh {\left (\frac {3 x}{2} \right )}}{4} + \frac {x \sinh {\left (\frac {3 x}{2} \right )} \sinh {\left (\frac {5 x}{2} \right )} \cosh {\left (x \right )}}{4} - \frac {x \cosh {\left (x \right )} \cosh {\left (\frac {3 x}{2} \right )} \cosh {\left (\frac {5 x}{2} \right )}}{4} - \frac {3 \sinh {\left (x \right )} \sinh {\left (\frac {3 x}{2} \right )} \sinh {\left (\frac {5 x}{2} \right )}}{20} + \frac {5 \sinh {\left (x \right )} \cosh {\left (\frac {3 x}{2} \right )} \cosh {\left (\frac {5 x}{2} \right )}}{12} - \frac {\sinh {\left (\frac {5 x}{2} \right )} \cosh {\left (x \right )} \cosh {\left (\frac {3 x}{2} \right )}}{15} \]
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none
Time = 0.27 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40 \[ \int \cosh \left (\frac {3 x}{2}\right ) \sinh (x) \sinh \left (\frac {5 x}{2}\right ) \, dx=-\frac {1}{240} \, {\left (10 \, e^{\left (-2 \, x\right )} - 15 \, e^{\left (-3 \, x\right )} - 6\right )} e^{\left (5 \, x\right )} - \frac {1}{4} \, x - \frac {1}{16} \, e^{\left (-2 \, x\right )} + \frac {1}{24} \, e^{\left (-3 \, x\right )} - \frac {1}{40} \, e^{\left (-5 \, x\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (22) = 44\).
Time = 0.29 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.60 \[ \int \cosh \left (\frac {3 x}{2}\right ) \sinh (x) \sinh \left (\frac {5 x}{2}\right ) \, dx=\frac {1}{240} \, {\left (137 \, e^{\left (5 \, x\right )} - 15 \, e^{\left (3 \, x\right )} + 10 \, e^{\left (2 \, x\right )} - 6\right )} e^{\left (-5 \, x\right )} - \frac {1}{4} \, x + \frac {1}{40} \, e^{\left (5 \, x\right )} - \frac {1}{24} \, e^{\left (3 \, x\right )} + \frac {1}{16} \, e^{\left (2 \, x\right )} \]
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Time = 0.56 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.33 \[ \int \cosh \left (\frac {3 x}{2}\right ) \sinh (x) \sinh \left (\frac {5 x}{2}\right ) \, dx=\frac {{\mathrm {e}}^{2\,x}}{16}-\frac {{\mathrm {e}}^{-2\,x}}{16}-\frac {x}{4}+\frac {{\mathrm {e}}^{-3\,x}}{24}-\frac {{\mathrm {e}}^{3\,x}}{24}-\frac {{\mathrm {e}}^{-5\,x}}{40}+\frac {{\mathrm {e}}^{5\,x}}{40} \]
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