Integrand size = 13, antiderivative size = 6 \[ \int \frac {e^x}{-1+e^{2 x}} \, dx=-\text {arctanh}\left (e^x\right ) \]
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Time = 0.01 (sec) , antiderivative size = 6, 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 = {2281, 213} \[ \int \frac {e^x}{-1+e^{2 x}} \, dx=-\text {arctanh}\left (e^x\right ) \]
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Rule 213
Rule 2281
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,e^x\right ) \\ & = -\tanh ^{-1}\left (e^x\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00 \[ \int \frac {e^x}{-1+e^{2 x}} \, dx=-\text {arctanh}\left (e^x\right ) \]
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Time = 0.02 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00
method | result | size |
default | \(-\operatorname {arctanh}\left ({\mathrm e}^{x}\right )\) | \(6\) |
norman | \(\frac {\ln \left (-1+{\mathrm e}^{x}\right )}{2}-\frac {\ln \left (1+{\mathrm e}^{x}\right )}{2}\) | \(16\) |
risch | \(\frac {\ln \left (-1+{\mathrm e}^{x}\right )}{2}-\frac {\ln \left (1+{\mathrm e}^{x}\right )}{2}\) | \(16\) |
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Leaf count of result is larger than twice the leaf count of optimal. 15 vs. \(2 (5) = 10\).
Time = 0.30 (sec) , antiderivative size = 15, normalized size of antiderivative = 2.50 \[ \int \frac {e^x}{-1+e^{2 x}} \, dx=-\frac {1}{2} \, \log \left (e^{x} + 1\right ) + \frac {1}{2} \, \log \left (e^{x} - 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 15 vs. \(2 (5) = 10\).
Time = 0.05 (sec) , antiderivative size = 15, normalized size of antiderivative = 2.50 \[ \int \frac {e^x}{-1+e^{2 x}} \, dx=\frac {\log {\left (e^{x} - 1 \right )}}{2} - \frac {\log {\left (e^{x} + 1 \right )}}{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 15 vs. \(2 (5) = 10\).
Time = 0.19 (sec) , antiderivative size = 15, normalized size of antiderivative = 2.50 \[ \int \frac {e^x}{-1+e^{2 x}} \, dx=-\frac {1}{2} \, \log \left (e^{x} + 1\right ) + \frac {1}{2} \, \log \left (e^{x} - 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 16 vs. \(2 (5) = 10\).
Time = 0.30 (sec) , antiderivative size = 16, normalized size of antiderivative = 2.67 \[ \int \frac {e^x}{-1+e^{2 x}} \, dx=-\frac {1}{2} \, \log \left (e^{x} + 1\right ) + \frac {1}{2} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]
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Time = 0.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 2.50 \[ \int \frac {e^x}{-1+e^{2 x}} \, dx=\frac {\ln \left ({\mathrm {e}}^x-1\right )}{2}-\frac {\ln \left ({\mathrm {e}}^x+1\right )}{2} \]
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