Integrand size = 16, antiderivative size = 38 \[ \int \frac {\sinh ^3(x)}{\cosh ^3(x)+\sinh ^3(x)} \, dx=\frac {x}{2}+\frac {2 \arctan \left (\frac {1-2 \tanh (x)}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{6 (1+\tanh (x))} \]
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Time = 0.11 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2099, 213, 632, 210} \[ \int \frac {\sinh ^3(x)}{\cosh ^3(x)+\sinh ^3(x)} \, dx=\frac {2 \arctan \left (\frac {1-2 \tanh (x)}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {x}{2}+\frac {1}{6 (\tanh (x)+1)} \]
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Rule 210
Rule 213
Rule 632
Rule 2099
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {x^3}{-1+x^2-x^3+x^5} \, dx,x,\tanh (x)\right ) \\ & = -\text {Subst}\left (\int \left (\frac {1}{6 (1+x)^2}+\frac {1}{2 \left (-1+x^2\right )}+\frac {1}{3 \left (1-x+x^2\right )}\right ) \, dx,x,\tanh (x)\right ) \\ & = \frac {1}{6 (1+\tanh (x))}-\frac {1}{3} \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\tanh (x)\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\tanh (x)\right ) \\ & = \frac {x}{2}+\frac {1}{6 (1+\tanh (x))}+\frac {2}{3} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \tanh (x)\right ) \\ & = \frac {x}{2}+\frac {2 \arctan \left (\frac {1-2 \tanh (x)}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{6 (1+\tanh (x))} \\ \end{align*}
Time = 5.04 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.05 \[ \int \frac {\sinh ^3(x)}{\cosh ^3(x)+\sinh ^3(x)} \, dx=\frac {1}{36} \left (18 x-8 \sqrt {3} \arctan \left (\frac {-1+2 \tanh (x)}{\sqrt {3}}\right )+3 \cosh (2 x)-3 \sinh (2 x)\right ) \]
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Result contains complex when optimal does not.
Time = 0.17 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.24
method | result | size |
risch | \(\frac {x}{2}+\frac {{\mathrm e}^{-2 x}}{12}+\frac {i \sqrt {3}\, \ln \left ({\mathrm e}^{2 x}-i \sqrt {3}\right )}{9}-\frac {i \sqrt {3}\, \ln \left ({\mathrm e}^{2 x}+i \sqrt {3}\right )}{9}\) | \(47\) |
default | \(\frac {i \sqrt {3}\, \ln \left (\tanh \left (\frac {x}{2}\right )^{2}+\left (i \sqrt {3}-1\right ) \tanh \left (\frac {x}{2}\right )+1\right )}{9}-\frac {i \sqrt {3}\, \ln \left (\tanh \left (\frac {x}{2}\right )^{2}+\left (-i \sqrt {3}-1\right ) \tanh \left (\frac {x}{2}\right )+1\right )}{9}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2}+\frac {1}{3 \left (1+\tanh \left (\frac {x}{2}\right )\right )^{2}}-\frac {1}{3 \left (1+\tanh \left (\frac {x}{2}\right )\right )}+\frac {\ln \left (1+\tanh \left (\frac {x}{2}\right )\right )}{2}\) | \(96\) |
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Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (29) = 58\).
Time = 0.28 (sec) , antiderivative size = 95, normalized size of antiderivative = 2.50 \[ \int \frac {\sinh ^3(x)}{\cosh ^3(x)+\sinh ^3(x)} \, dx=\frac {18 \, x \cosh \left (x\right )^{2} + 36 \, x \cosh \left (x\right ) \sinh \left (x\right ) + 18 \, x \sinh \left (x\right )^{2} + 8 \, {\left (\sqrt {3} \cosh \left (x\right )^{2} + 2 \, \sqrt {3} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt {3} \sinh \left (x\right )^{2}\right )} \arctan \left (-\frac {\sqrt {3} \cosh \left (x\right ) + \sqrt {3} \sinh \left (x\right )}{3 \, {\left (\cosh \left (x\right ) - \sinh \left (x\right )\right )}}\right ) + 3}{36 \, {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (36) = 72\).
Time = 0.50 (sec) , antiderivative size = 136, normalized size of antiderivative = 3.58 \[ \int \frac {\sinh ^3(x)}{\cosh ^3(x)+\sinh ^3(x)} \, dx=\frac {9 x \sinh {\left (x \right )}}{18 \sinh {\left (x \right )} + 18 \cosh {\left (x \right )}} + \frac {9 x \cosh {\left (x \right )}}{18 \sinh {\left (x \right )} + 18 \cosh {\left (x \right )}} - \frac {4 \sqrt {3} \sinh {\left (x \right )} \operatorname {atan}{\left (\frac {\sqrt {3}}{3} - \frac {2 \sqrt {3} \cosh {\left (x \right )}}{3 \sinh {\left (x \right )}} \right )}}{18 \sinh {\left (x \right )} + 18 \cosh {\left (x \right )}} - \frac {4 \sqrt {3} \cosh {\left (x \right )} \operatorname {atan}{\left (\frac {\sqrt {3}}{3} - \frac {2 \sqrt {3} \cosh {\left (x \right )}}{3 \sinh {\left (x \right )}} \right )}}{18 \sinh {\left (x \right )} + 18 \cosh {\left (x \right )}} + \frac {3 \cosh {\left (x \right )}}{18 \sinh {\left (x \right )} + 18 \cosh {\left (x \right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (29) = 58\).
Time = 0.29 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.92 \[ \int \frac {\sinh ^3(x)}{\cosh ^3(x)+\sinh ^3(x)} \, dx=-\frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (2 \, \sqrt {3} e^{\left (-x\right )} + 3^{\frac {1}{4}} \sqrt {2}\right )}\right ) + \frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (2 \, \sqrt {3} e^{\left (-x\right )} - 3^{\frac {1}{4}} \sqrt {2}\right )}\right ) + \frac {1}{2} \, x + \frac {1}{12} \, e^{\left (-2 \, x\right )} \]
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none
Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.87 \[ \int \frac {\sinh ^3(x)}{\cosh ^3(x)+\sinh ^3(x)} \, dx=-\frac {1}{12} \, {\left (3 \, e^{\left (2 \, x\right )} - 1\right )} e^{\left (-2 \, x\right )} - \frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} e^{\left (2 \, x\right )}\right ) + \frac {1}{2} \, x \]
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Time = 2.48 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.66 \[ \int \frac {\sinh ^3(x)}{\cosh ^3(x)+\sinh ^3(x)} \, dx=\frac {x}{2}+\frac {{\mathrm {e}}^{-2\,x}}{12}-\frac {2\,\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,{\mathrm {e}}^{2\,x}}{3}\right )}{9} \]
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