Integrand size = 14, antiderivative size = 13 \[ \int \frac {e^x (1+\sinh (x))}{1+\cosh (x)} \, dx=e^x+\frac {2}{1+e^x} \]
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Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2320, 697} \[ \int \frac {e^x (1+\sinh (x))}{1+\cosh (x)} \, dx=e^x+\frac {2}{e^x+1} \]
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Rule 697
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {-1+2 x+x^2}{(1+x)^2} \, dx,x,e^x\right ) \\ & = \text {Subst}\left (\int \left (1-\frac {2}{(1+x)^2}\right ) \, dx,x,e^x\right ) \\ & = e^x+\frac {2}{1+e^x} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.38 \[ \int \frac {e^x (1+\sinh (x))}{1+\cosh (x)} \, dx=\frac {2+e^x+e^{2 x}}{1+e^x} \]
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Time = 0.19 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92
method | result | size |
risch | \({\mathrm e}^{x}+\frac {2}{1+{\mathrm e}^{x}}\) | \(12\) |
default | \(-\tanh \left (\frac {x}{2}\right )-\frac {2}{\tanh \left (\frac {x}{2}\right )-1}\) | \(18\) |
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none
Time = 0.23 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.62 \[ \int \frac {e^x (1+\sinh (x))}{1+\cosh (x)} \, dx=\frac {3 \, \cosh \left (x\right ) - \sinh \left (x\right ) + 1}{\cosh \left (x\right ) - \sinh \left (x\right ) + 1} \]
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\[ \int \frac {e^x (1+\sinh (x))}{1+\cosh (x)} \, dx=\int \frac {\left (\sinh {\left (x \right )} + 1\right ) e^{x}}{\cosh {\left (x \right )} + 1}\, dx \]
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none
Time = 0.20 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {e^x (1+\sinh (x))}{1+\cosh (x)} \, dx=\frac {2}{e^{x} + 1} + e^{x} \]
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none
Time = 0.28 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {e^x (1+\sinh (x))}{1+\cosh (x)} \, dx=\frac {2}{e^{x} + 1} + e^{x} \]
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Time = 0.35 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {e^x (1+\sinh (x))}{1+\cosh (x)} \, dx={\mathrm {e}}^x+\frac {2}{{\mathrm {e}}^x+1} \]
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