Integrand size = 13, antiderivative size = 9 \[ \int \frac {e^x}{\cosh (x)-\sinh (x)} \, dx=\frac {e^{2 x}}{2} \]
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Time = 0.01 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2320, 30} \[ \int \frac {e^x}{\cosh (x)-\sinh (x)} \, dx=\frac {e^{2 x}}{2} \]
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Rule 30
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int x \, dx,x,e^x\right ) \\ & = \frac {e^{2 x}}{2} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \frac {e^x}{\cosh (x)-\sinh (x)} \, dx=\frac {e^{2 x}}{2} \]
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Time = 0.60 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78
method | result | size |
risch | \(\frac {{\mathrm e}^{2 x}}{2}\) | \(7\) |
gosper | \(\frac {{\mathrm e}^{x}}{2 \cosh \left (x \right )-2 \sinh \left (x \right )}\) | \(14\) |
default | \(\frac {2}{\tanh \left (\frac {x}{2}\right )-1}+\frac {2}{\left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}\) | \(22\) |
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Leaf count of result is larger than twice the leaf count of optimal. 16 vs. \(2 (6) = 12\).
Time = 0.23 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.78 \[ \int \frac {e^x}{\cosh (x)-\sinh (x)} \, dx=\frac {\cosh \left (x\right ) + \sinh \left (x\right )}{2 \, {\left (\cosh \left (x\right ) - \sinh \left (x\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 12 vs. \(2 (5) = 10\).
Time = 0.21 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.33 \[ \int \frac {e^x}{\cosh (x)-\sinh (x)} \, dx=\frac {e^{x}}{- 2 \sinh {\left (x \right )} + 2 \cosh {\left (x \right )}} \]
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none
Time = 0.19 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.67 \[ \int \frac {e^x}{\cosh (x)-\sinh (x)} \, dx=\frac {1}{2} \, e^{\left (2 \, x\right )} \]
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none
Time = 0.28 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.67 \[ \int \frac {e^x}{\cosh (x)-\sinh (x)} \, dx=\frac {1}{2} \, e^{\left (2 \, x\right )} \]
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Time = 0.37 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.67 \[ \int \frac {e^x}{\cosh (x)-\sinh (x)} \, dx=\frac {{\mathrm {e}}^{2\,x}}{2} \]
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