Optimal. Leaf size=24 \[ -\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{5}} e^{m x}\right )}{\sqrt{10} m} \]
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Rubi [A] time = 0.0178917, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2282, 207} \[ -\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{5}} e^{m x}\right )}{\sqrt{10} m} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 207
Rubi steps
\begin{align*} \int \frac{1}{-5 e^{-m x}+2 e^{m x}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{-5+2 x^2} \, dx,x,e^{m x}\right )}{m}\\ &=-\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{5}} e^{m x}\right )}{\sqrt{10} m}\\ \end{align*}
Mathematica [A] time = 0.0096272, size = 24, normalized size = 1. \[ -\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{5}} e^{m x}\right )}{\sqrt{10} m} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 19, normalized size = 0.8 \begin{align*} -{\frac{\sqrt{10}}{10\,m}{\it Artanh} \left ({\frac{{{\rm e}^{mx}}\sqrt{10}}{5}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.40219, size = 47, normalized size = 1.96 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.92609, size = 108, normalized size = 4.5 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.13892, size = 20, normalized size = 0.83 \begin{align*} \frac{\operatorname{RootSum}{\left (40 z^{2} - 1, \left ( i \mapsto i \log{\left (- 4 i + e^{- m x} \right )} \right )\right )}}{m} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.10253, size = 51, normalized size = 2.12 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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