Integrand size = 11, antiderivative size = 25 \[ \int \frac {\sinh ^3(x)}{1+\tanh (x)} \, dx=-\frac {1}{3} \cosh ^3(x)+\frac {\cosh ^5(x)}{5}-\frac {\sinh ^5(x)}{5} \]
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Time = 0.14 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {3599, 3187, 3186, 2645, 14, 2644, 30} \[ \int \frac {\sinh ^3(x)}{1+\tanh (x)} \, dx=-\frac {\sinh ^5(x)}{5}+\frac {\cosh ^5(x)}{5}-\frac {\cosh ^3(x)}{3} \]
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Rule 14
Rule 30
Rule 2644
Rule 2645
Rule 3186
Rule 3187
Rule 3599
Rubi steps \begin{align*} \text {integral}& = \int \frac {\cosh (x) \sinh ^3(x)}{\cosh (x)+\sinh (x)} \, dx \\ & = i \int \cosh (x) (-i \cosh (x)+i \sinh (x)) \sinh ^3(x) \, dx \\ & = -\int \left (-\cosh ^2(x) \sinh ^3(x)+\cosh (x) \sinh ^4(x)\right ) \, dx \\ & = \int \cosh ^2(x) \sinh ^3(x) \, dx-\int \cosh (x) \sinh ^4(x) \, dx \\ & = i \text {Subst}\left (\int x^4 \, dx,x,i \sinh (x)\right )-\text {Subst}\left (\int x^2 \left (1-x^2\right ) \, dx,x,\cosh (x)\right ) \\ & = -\frac {1}{5} \sinh ^5(x)-\text {Subst}\left (\int \left (x^2-x^4\right ) \, dx,x,\cosh (x)\right ) \\ & = -\frac {1}{3} \cosh ^3(x)+\frac {\cosh ^5(x)}{5}-\frac {\sinh ^5(x)}{5} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \frac {\sinh ^3(x)}{1+\tanh (x)} \, dx=\frac {1}{120} (\cosh (x)-\sinh (x)) (-20 \cosh (2 x)+4 \cosh (4 x)-10 \sinh (2 x)+\sinh (4 x)) \]
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Time = 0.55 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96
method | result | size |
risch | \(\frac {{\mathrm e}^{3 x}}{48}-\frac {{\mathrm e}^{x}}{8}-\frac {{\mathrm e}^{-3 x}}{24}+\frac {{\mathrm e}^{-5 x}}{80}\) | \(24\) |
parallelrisch | \(-\frac {2}{15}-\frac {\cosh \left (x \right )}{8}-\frac {\sinh \left (x \right )}{8}-\frac {\sinh \left (5 x \right )}{80}-\frac {\cosh \left (3 x \right )}{48}+\frac {\cosh \left (5 x \right )}{80}+\frac {\sinh \left (3 x \right )}{16}\) | \(35\) |
default | \(\frac {2}{5 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{5}}-\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}+\frac {2}{3 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {1}{8 \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {1}{6 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {1}{4 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {1}{8 \tanh \left (\frac {x}{2}\right )-8}\) | \(72\) |
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Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (19) = 38\).
Time = 0.25 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.24 \[ \int \frac {\sinh ^3(x)}{1+\tanh (x)} \, dx=\frac {\cosh \left (x\right )^{4} + \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + {\left (6 \, \cosh \left (x\right )^{2} - 5\right )} \sinh \left (x\right )^{2} - 5 \, \cosh \left (x\right )^{2} + {\left (\cosh \left (x\right )^{3} - 5 \, \cosh \left (x\right )\right )} \sinh \left (x\right )}{30 \, {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (19) = 38\).
Time = 0.34 (sec) , antiderivative size = 134, normalized size of antiderivative = 5.36 \[ \int \frac {\sinh ^3(x)}{1+\tanh (x)} \, dx=\frac {3 \sinh ^{3}{\left (x \right )} \tanh {\left (x \right )}}{15 \tanh {\left (x \right )} + 15} - \frac {3 \sinh ^{3}{\left (x \right )}}{15 \tanh {\left (x \right )} + 15} + \frac {6 \sinh ^{2}{\left (x \right )} \cosh {\left (x \right )} \tanh {\left (x \right )}}{15 \tanh {\left (x \right )} + 15} + \frac {9 \sinh ^{2}{\left (x \right )} \cosh {\left (x \right )}}{15 \tanh {\left (x \right )} + 15} - \frac {6 \sinh {\left (x \right )} \cosh ^{2}{\left (x \right )} \tanh {\left (x \right )}}{15 \tanh {\left (x \right )} + 15} + \frac {6 \sinh {\left (x \right )} \cosh ^{2}{\left (x \right )}}{15 \tanh {\left (x \right )} + 15} - \frac {8 \cosh ^{3}{\left (x \right )} \tanh {\left (x \right )}}{15 \tanh {\left (x \right )} + 15} - \frac {2 \cosh ^{3}{\left (x \right )}}{15 \tanh {\left (x \right )} + 15} \]
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Time = 0.19 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {\sinh ^3(x)}{1+\tanh (x)} \, dx=-\frac {1}{48} \, {\left (6 \, e^{\left (-2 \, x\right )} - 1\right )} e^{\left (3 \, x\right )} - \frac {1}{24} \, e^{\left (-3 \, x\right )} + \frac {1}{80} \, e^{\left (-5 \, x\right )} \]
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Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {\sinh ^3(x)}{1+\tanh (x)} \, dx=-\frac {1}{240} \, {\left (10 \, e^{\left (2 \, x\right )} - 3\right )} e^{\left (-5 \, x\right )} + \frac {1}{48} \, e^{\left (3 \, x\right )} - \frac {1}{8} \, e^{x} \]
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Time = 1.77 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {\sinh ^3(x)}{1+\tanh (x)} \, dx=\frac {{\mathrm {e}}^{3\,x}}{48}-\frac {{\mathrm {e}}^{-3\,x}}{24}+\frac {{\mathrm {e}}^{-5\,x}}{80}-\frac {{\mathrm {e}}^x}{8} \]
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