Integrand size = 18, antiderivative size = 20 \[ \int \frac {e^x}{-1-8 e^x+e^{2 x}} \, dx=\frac {\text {arctanh}\left (\frac {4-e^x}{\sqrt {17}}\right )}{\sqrt {17}} \]
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Time = 0.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2320, 632, 212} \[ \int \frac {e^x}{-1-8 e^x+e^{2 x}} \, dx=\frac {\text {arctanh}\left (\frac {4-e^x}{\sqrt {17}}\right )}{\sqrt {17}} \]
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Rule 212
Rule 632
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{-1-8 x+x^2} \, dx,x,e^x\right ) \\ & = -\left (2 \text {Subst}\left (\int \frac {1}{68-x^2} \, dx,x,-8+2 e^x\right )\right ) \\ & = \frac {\tanh ^{-1}\left (\frac {4-e^x}{\sqrt {17}}\right )}{\sqrt {17}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {e^x}{-1-8 e^x+e^{2 x}} \, dx=-\frac {\text {arctanh}\left (\frac {-4+e^x}{\sqrt {17}}\right )}{\sqrt {17}} \]
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Time = 0.04 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90
method | result | size |
default | \(-\frac {\sqrt {17}\, \operatorname {arctanh}\left (\frac {\left (2 \,{\mathrm e}^{x}-8\right ) \sqrt {17}}{34}\right )}{17}\) | \(18\) |
risch | \(\frac {\sqrt {17}\, \ln \left ({\mathrm e}^{x}-4-\sqrt {17}\right )}{34}-\frac {\sqrt {17}\, \ln \left ({\mathrm e}^{x}-4+\sqrt {17}\right )}{34}\) | \(30\) |
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Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (15) = 30\).
Time = 0.29 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.10 \[ \int \frac {e^x}{-1-8 e^x+e^{2 x}} \, dx=\frac {1}{34} \, \sqrt {17} \log \left (-\frac {2 \, {\left (\sqrt {17} + 4\right )} e^{x} - 8 \, \sqrt {17} - e^{\left (2 \, x\right )} - 33}{e^{\left (2 \, x\right )} - 8 \, e^{x} - 1}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {e^x}{-1-8 e^x+e^{2 x}} \, dx=\operatorname {RootSum} {\left (68 z^{2} - 1, \left ( i \mapsto i \log {\left (- 34 i + e^{x} - 4 \right )} \right )\right )} \]
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none
Time = 0.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.30 \[ \int \frac {e^x}{-1-8 e^x+e^{2 x}} \, dx=\frac {1}{34} \, \sqrt {17} \log \left (-\frac {\sqrt {17} - e^{x} + 4}{\sqrt {17} + e^{x} - 4}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (15) = 30\).
Time = 0.29 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.65 \[ \int \frac {e^x}{-1-8 e^x+e^{2 x}} \, dx=\frac {1}{34} \, \sqrt {17} \log \left (\frac {{\left | -2 \, \sqrt {17} + 2 \, e^{x} - 8 \right |}}{{\left | 2 \, \sqrt {17} + 2 \, e^{x} - 8 \right |}}\right ) \]
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Time = 0.55 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {e^x}{-1-8 e^x+e^{2 x}} \, dx=-\frac {\sqrt {17}\,\mathrm {atanh}\left (\frac {\sqrt {17}\,\left (2\,{\mathrm {e}}^x-8\right )}{34}\right )}{17} \]
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