Integrand size = 18, antiderivative size = 24 \[ \int \frac {1}{-5 e^{-m x}+2 e^{m x}} \, dx=-\frac {\text {arctanh}\left (\sqrt {\frac {2}{5}} e^{m x}\right )}{\sqrt {10} m} \]
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Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2320, 213} \[ \int \frac {1}{-5 e^{-m x}+2 e^{m x}} \, dx=-\frac {\text {arctanh}\left (\sqrt {\frac {2}{5}} e^{m x}\right )}{\sqrt {10} m} \]
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Rule 213
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{-5+2 x^2} \, dx,x,e^{m x}\right )}{m} \\ & = -\frac {\text {arctanh}\left (\sqrt {\frac {2}{5}} e^{m x}\right )}{\sqrt {10} m} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{-5 e^{-m x}+2 e^{m x}} \, dx=-\frac {\text {arctanh}\left (\sqrt {\frac {2}{5}} e^{m x}\right )}{\sqrt {10} m} \]
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Time = 0.05 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79
method | result | size |
derivativedivides | \(-\frac {\operatorname {arctanh}\left (\frac {{\mathrm e}^{m x} \sqrt {10}}{5}\right ) \sqrt {10}}{10 m}\) | \(19\) |
default | \(-\frac {\operatorname {arctanh}\left (\frac {{\mathrm e}^{m x} \sqrt {10}}{5}\right ) \sqrt {10}}{10 m}\) | \(19\) |
risch | \(\frac {\sqrt {10}\, \ln \left ({\mathrm e}^{m x}-\frac {\sqrt {10}}{2}\right )}{20 m}-\frac {\sqrt {10}\, \ln \left ({\mathrm e}^{m x}+\frac {\sqrt {10}}{2}\right )}{20 m}\) | \(40\) |
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Leaf count of result is larger than twice the leaf count of optimal. 40 vs. \(2 (18) = 36\).
Time = 0.23 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.67 \[ \int \frac {1}{-5 e^{-m x}+2 e^{m x}} \, dx=\frac {\sqrt {10} \log \left (-\frac {2 \, \sqrt {10} e^{\left (m x\right )} - 2 \, e^{\left (2 \, m x\right )} - 5}{2 \, e^{\left (2 \, m x\right )} - 5}\right )}{20 \, m} \]
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Time = 0.06 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {1}{-5 e^{-m x}+2 e^{m x}} \, dx=\frac {\operatorname {RootSum} {\left (40 z^{2} - 1, \left ( i \mapsto i \log {\left (- 4 i + e^{- m x} \right )} \right )\right )}}{m} \]
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none
Time = 0.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.46 \[ \int \frac {1}{-5 e^{-m x}+2 e^{m x}} \, dx=\frac {\sqrt {10} \log \left (-\frac {\sqrt {10} - 5 \, e^{\left (-m x\right )}}{\sqrt {10} + 5 \, e^{\left (-m x\right )}}\right )}{20 \, m} \]
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Leaf count of result is larger than twice the leaf count of optimal. 38 vs. \(2 (18) = 36\).
Time = 0.29 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.58 \[ \int \frac {1}{-5 e^{-m x}+2 e^{m x}} \, dx=-\frac {\sqrt {10} \log \left (\frac {1}{2} \, \sqrt {10} + e^{\left (m x\right )}\right ) - \sqrt {10} \log \left ({\left | -\frac {1}{2} \, \sqrt {10} + e^{\left (m x\right )} \right |}\right )}{20 \, m} \]
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Time = 0.52 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.75 \[ \int \frac {1}{-5 e^{-m x}+2 e^{m x}} \, dx=-\frac {\sqrt {10}\,\mathrm {atanh}\left (\frac {\sqrt {10}\,{\mathrm {e}}^{m\,x}}{5}\right )}{10\,m} \]
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