Integrand size = 12, antiderivative size = 22 \[ \int \frac {e^x}{1-\cosh (x)} \, dx=-\frac {2}{1-e^x}-2 \log \left (1-e^x\right ) \]
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Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2320, 12, 45} \[ \int \frac {e^x}{1-\cosh (x)} \, dx=-\frac {2}{1-e^x}-2 \log \left (1-e^x\right ) \]
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Rule 12
Rule 45
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int -\frac {2 x}{(1-x)^2} \, dx,x,e^x\right ) \\ & = -\left (2 \text {Subst}\left (\int \frac {x}{(1-x)^2} \, dx,x,e^x\right )\right ) \\ & = -\left (2 \text {Subst}\left (\int \left (\frac {1}{(-1+x)^2}+\frac {1}{-1+x}\right ) \, dx,x,e^x\right )\right ) \\ & = -\frac {2}{1-e^x}-2 \log \left (1-e^x\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.64 \[ \int \frac {e^x}{1-\cosh (x)} \, dx=\frac {4 \left (\frac {1}{1-e^x}+\log \left (1-e^x\right )\right ) \sinh ^2\left (\frac {x}{2}\right )}{1-\cosh (x)} \]
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Time = 0.09 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77
method | result | size |
risch | \(\frac {2}{-1+{\mathrm e}^{x}}-2 \ln \left (-1+{\mathrm e}^{x}\right )\) | \(17\) |
default | \(2 \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+\frac {1}{\tanh \left (\frac {x}{2}\right )}-2 \ln \left (\tanh \left (\frac {x}{2}\right )\right )\) | \(24\) |
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Time = 0.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {e^x}{1-\cosh (x)} \, dx=-\frac {2 \, {\left ({\left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) - 1\right )}}{\cosh \left (x\right ) + \sinh \left (x\right ) - 1} \]
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\[ \int \frac {e^x}{1-\cosh (x)} \, dx=- \int \frac {e^{x}}{\cosh {\left (x \right )} - 1}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.73 \[ \int \frac {e^x}{1-\cosh (x)} \, dx=\frac {2}{e^{x} - 1} - 2 \, \log \left (e^{x} - 1\right ) \]
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Time = 0.28 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int \frac {e^x}{1-\cosh (x)} \, dx=\frac {2}{e^{x} - 1} - 2 \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.73 \[ \int \frac {e^x}{1-\cosh (x)} \, dx=\frac {2}{{\mathrm {e}}^x-1}-2\,\ln \left ({\mathrm {e}}^x-1\right ) \]
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