Integrand size = 15, antiderivative size = 12 \[ \int \frac {e^x}{16-e^{2 x}} \, dx=\frac {1}{4} \text {arctanh}\left (\frac {e^x}{4}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2281, 212} \[ \int \frac {e^x}{16-e^{2 x}} \, dx=\frac {1}{4} \text {arctanh}\left (\frac {e^x}{4}\right ) \]
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Rule 212
Rule 2281
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{16-x^2} \, dx,x,e^x\right ) \\ & = \frac {1}{4} \tanh ^{-1}\left (\frac {e^x}{4}\right ) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {e^x}{16-e^{2 x}} \, dx=\frac {1}{4} \text {arctanh}\left (\frac {e^x}{4}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(15\) vs. \(2(7)=14\).
Time = 0.03 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.33
method | result | size |
default | \(\frac {\ln \left ({\mathrm e}^{x}+4\right )}{8}-\frac {\ln \left ({\mathrm e}^{x}-4\right )}{8}\) | \(16\) |
norman | \(\frac {\ln \left ({\mathrm e}^{x}+4\right )}{8}-\frac {\ln \left ({\mathrm e}^{x}-4\right )}{8}\) | \(16\) |
risch | \(\frac {\ln \left ({\mathrm e}^{x}+4\right )}{8}-\frac {\ln \left ({\mathrm e}^{x}-4\right )}{8}\) | \(16\) |
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Leaf count of result is larger than twice the leaf count of optimal. 15 vs. \(2 (7) = 14\).
Time = 0.30 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.25 \[ \int \frac {e^x}{16-e^{2 x}} \, dx=\frac {1}{8} \, \log \left (e^{x} + 4\right ) - \frac {1}{8} \, \log \left (e^{x} - 4\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 15 vs. \(2 (7) = 14\).
Time = 0.05 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.25 \[ \int \frac {e^x}{16-e^{2 x}} \, dx=- \frac {\log {\left (e^{x} - 4 \right )}}{8} + \frac {\log {\left (e^{x} + 4 \right )}}{8} \]
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Leaf count of result is larger than twice the leaf count of optimal. 15 vs. \(2 (7) = 14\).
Time = 0.20 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.25 \[ \int \frac {e^x}{16-e^{2 x}} \, dx=\frac {1}{8} \, \log \left (e^{x} + 4\right ) - \frac {1}{8} \, \log \left (e^{x} - 4\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 16 vs. \(2 (7) = 14\).
Time = 0.31 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.33 \[ \int \frac {e^x}{16-e^{2 x}} \, dx=\frac {1}{8} \, \log \left (e^{x} + 4\right ) - \frac {1}{8} \, \log \left ({\left | e^{x} - 4 \right |}\right ) \]
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Time = 0.33 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.25 \[ \int \frac {e^x}{16-e^{2 x}} \, dx=\frac {\ln \left ({\mathrm {e}}^x+4\right )}{8}-\frac {\ln \left ({\mathrm {e}}^x-4\right )}{8} \]
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