Integrand size = 13, antiderivative size = 13 \[ \int \frac {e^{m x}}{\cosh (x)+\sinh (x)} \, dx=\frac {e^{(-1+m) x}}{-1+m} \]
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Time = 0.03 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.46, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {5767, 2259, 2225} \[ \int \frac {e^{m x}}{\cosh (x)+\sinh (x)} \, dx=-\frac {e^{-((1-m) x)}}{1-m} \]
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Rule 2225
Rule 2259
Rule 5767
Rubi steps \begin{align*} \text {integral}& = \int e^{-x+m x} \, dx \\ & = \int e^{-((1-m) x)} \, dx \\ & = -\frac {e^{-((1-m) x)}}{1-m} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.38 \[ \int \frac {e^{m x}}{\cosh (x)+\sinh (x)} \, dx=\frac {e^{m x} (\cosh (x)-\sinh (x))}{-1+m} \]
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Time = 0.56 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00
method | result | size |
risch | \(\frac {{\mathrm e}^{\left (-1+m \right ) x}}{-1+m}\) | \(13\) |
gosper | \(\frac {{\mathrm e}^{m x}}{\left (-1+m \right ) \left (\cosh \left (x \right )+\sinh \left (x \right )\right )}\) | \(18\) |
default | \(\frac {\sinh \left (\left (-1+m \right ) x \right )}{-1+m}+\frac {\cosh \left (\left (-1+m \right ) x \right )}{-1+m}\) | \(26\) |
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Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (12) = 24\).
Time = 0.24 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.92 \[ \int \frac {e^{m x}}{\cosh (x)+\sinh (x)} \, dx=\frac {\cosh \left (m x\right ) + \sinh \left (m x\right )}{{\left (m - 1\right )} \cosh \left (x\right ) + {\left (m - 1\right )} \sinh \left (x\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (8) = 16\).
Time = 0.23 (sec) , antiderivative size = 32, normalized size of antiderivative = 2.46 \[ \int \frac {e^{m x}}{\cosh (x)+\sinh (x)} \, dx=\begin {cases} \frac {e^{m x}}{m \sinh {\left (x \right )} + m \cosh {\left (x \right )} - \sinh {\left (x \right )} - \cosh {\left (x \right )}} & \text {for}\: m \neq 1 \\\frac {x e^{x}}{\sinh {\left (x \right )} + \cosh {\left (x \right )}} & \text {otherwise} \end {cases} \]
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Exception generated. \[ \int \frac {e^{m x}}{\cosh (x)+\sinh (x)} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.29 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.23 \[ \int \frac {e^{m x}}{\cosh (x)+\sinh (x)} \, dx=\frac {e^{\left (m x\right )}}{m e^{x} - e^{x}} \]
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Time = 0.11 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08 \[ \int \frac {e^{m x}}{\cosh (x)+\sinh (x)} \, dx=\frac {{\mathrm {e}}^{m\,x-x}}{m-1} \]
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