Optimal. Leaf size=31 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} e^{m x}}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b} m} \]
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Rubi [A] time = 0.0287608, antiderivative size = 31, 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, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} e^{m x}}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b} m} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{b e^{-m x}+a e^{m x}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{b+a x^2} \, dx,x,e^{m x}\right )}{m}\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{a} e^{m x}}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b} m}\\ \end{align*}
Mathematica [A] time = 0.0091206, size = 31, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} e^{m x}}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b} m} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 22, normalized size = 0.7 \begin{align*}{\frac{1}{m}\arctan \left ({a{{\rm e}^{mx}}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73, size = 192, normalized size = 6.19 \begin{align*} \left [-\frac{\sqrt{-a b} \log \left (\frac{a e^{\left (2 \, m x\right )} - 2 \, \sqrt{-a b} e^{\left (m x\right )} - b}{a e^{\left (2 \, m x\right )} + b}\right )}{2 \, a b m}, -\frac{\sqrt{a b} \arctan \left (\frac{\sqrt{a b} e^{\left (-m x\right )}}{a}\right )}{a b m}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.176954, size = 26, normalized size = 0.84 \begin{align*} \frac{\operatorname{RootSum}{\left (4 z^{2} a b + 1, \left ( i \mapsto i \log{\left (- 2 i a + e^{- m x} \right )} \right )\right )}}{m} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09318, size = 28, normalized size = 0.9 \begin{align*} \frac{\arctan \left (\frac{a e^{\left (m x\right )}}{\sqrt{a b}}\right )}{\sqrt{a b} m} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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