Integrand size = 11, antiderivative size = 25 \[ \int \frac {\cosh ^3(x)}{1+\coth (x)} \, dx=\frac {\cosh ^5(x)}{5}-\frac {\sinh ^3(x)}{3}-\frac {\sinh ^5(x)}{5} \]
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Time = 0.13 (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, 30, 2644, 14} \[ \int \frac {\cosh ^3(x)}{1+\coth (x)} \, dx=-\frac {\sinh ^5(x)}{5}-\frac {\sinh ^3(x)}{3}+\frac {\cosh ^5(x)}{5} \]
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Rule 14
Rule 30
Rule 2644
Rule 2645
Rule 3186
Rule 3187
Rule 3599
Rubi steps \begin{align*} \text {integral}& = -\left (i \int \frac {\cosh ^3(x) \sinh (x)}{-i \cosh (x)-i \sinh (x)} \, dx\right ) \\ & = -\int \cosh ^3(x) \sinh (x) (-\cosh (x)+\sinh (x)) \, dx \\ & = i \int \left (-i \cosh ^4(x) \sinh (x)+i \cosh ^3(x) \sinh ^2(x)\right ) \, dx \\ & = \int \cosh ^4(x) \sinh (x) \, dx-\int \cosh ^3(x) \sinh ^2(x) \, dx \\ & = -\left (i \text {Subst}\left (\int x^2 \left (1-x^2\right ) \, dx,x,i \sinh (x)\right )\right )+\text {Subst}\left (\int x^4 \, dx,x,\cosh (x)\right ) \\ & = \frac {\cosh ^5(x)}{5}-i \text {Subst}\left (\int \left (x^2-x^4\right ) \, dx,x,i \sinh (x)\right ) \\ & = \frac {\cosh ^5(x)}{5}-\frac {\sinh ^3(x)}{3}-\frac {\sinh ^5(x)}{5} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \frac {\cosh ^3(x)}{1+\coth (x)} \, dx=\frac {1}{120} (\cosh (x)-\sinh (x)) (20 \cosh (2 x)+4 \cosh (4 x)+10 \sinh (2 x)+\sinh (4 x)) \]
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Time = 0.56 (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 | \(\frac {\cosh \left (3 x \right )}{16}+\frac {\cosh \left (x \right )}{8}-\frac {\sinh \left (3 x \right )}{48}+\frac {\sinh \left (x \right )}{8}-\frac {2}{15}-\frac {\sinh \left (5 x \right )}{80}+\frac {\cosh \left (5 x \right )}{80}\) | \(35\) |
default | \(-\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}+\frac {2}{5 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{5}}+\frac {4}{3 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {3}{8 \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {1}{6 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {1}{4 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {3}{8 \left (\tanh \left (\frac {x}{2}\right )-1\right )}\) | \(82\) |
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Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (19) = 38\).
Time = 0.24 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.24 \[ \int \frac {\cosh ^3(x)}{1+\coth (x)} \, dx=\frac {\cosh \left (x\right )^{4} + \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + {\left (6 \, \cosh \left (x\right )^{2} + 5\right )} \sinh \left (x\right )^{2} + 5 \, \cosh \left (x\right )^{2} + {\left (\cosh \left (x\right )^{3} + 5 \, \cosh \left (x\right )\right )} \sinh \left (x\right )}{30 \, {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \]
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\[ \int \frac {\cosh ^3(x)}{1+\coth (x)} \, dx=\int \frac {\cosh ^{3}{\left (x \right )}}{\coth {\left (x \right )} + 1}\, dx \]
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none
Time = 0.19 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {\cosh ^3(x)}{1+\coth (x)} \, dx=\frac {1}{48} \, {\left (6 \, e^{\left (-2 \, x\right )} + 1\right )} e^{\left (3 \, x\right )} + \frac {1}{24} \, e^{\left (-3 \, x\right )} + \frac {1}{80} \, e^{\left (-5 \, x\right )} \]
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none
Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {\cosh ^3(x)}{1+\coth (x)} \, dx=\frac {1}{240} \, {\left (10 \, e^{\left (2 \, x\right )} + 3\right )} e^{\left (-5 \, x\right )} + \frac {1}{48} \, e^{\left (3 \, x\right )} + \frac {1}{8} \, e^{x} \]
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Time = 1.99 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {\cosh ^3(x)}{1+\coth (x)} \, dx=\frac {{\mathrm {e}}^{-3\,x}}{24}+\frac {{\mathrm {e}}^{3\,x}}{48}+\frac {{\mathrm {e}}^{-5\,x}}{80}+\frac {{\mathrm {e}}^x}{8} \]
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