Optimal. Leaf size=31 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} e^{m x}}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b} m} \]
[Out]
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Rubi [A] time = 0.0514737, 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 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} e^{m x}}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b} m} \]
Antiderivative was successfully verified.
[In] Int[(b/E^(m*x) + a*E^(m*x))^(-1),x]
[Out]
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Rubi in Sympy [A] time = 5.50959, size = 29, normalized size = 0.94 \[ - \frac{\operatorname{atan}{\left (\frac{\sqrt{b} e^{- m x}}{\sqrt{a}} \right )}}{\sqrt{a} \sqrt{b} m} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b/exp(m*x)+a*exp(m*x)),x)
[Out]
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Mathematica [A] time = 0.0136105, 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.
[In] Integrate[(b/E^(m*x) + a*E^(m*x))^(-1),x]
[Out]
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Maple [A] time = 0.006, size = 22, normalized size = 0.7 \[{\frac{1}{m}\arctan \left ({a{{\rm e}^{mx}}{\frac{1}{\sqrt{ba}}}} \right ){\frac{1}{\sqrt{ba}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b/exp(m*x)+a*exp(m*x)),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a*e^(m*x) + b*e^(-m*x)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.222215, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (\frac{2 \, a b e^{\left (m x\right )} + \sqrt{-a b}{\left (a e^{\left (2 \, m x\right )} - b\right )}}{a e^{\left (2 \, m x\right )} + b}\right )}{2 \, \sqrt{-a b} m}, -\frac{\arctan \left (\frac{b e^{\left (-m x\right )}}{\sqrt{a b}}\right )}{\sqrt{a b} m}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a*e^(m*x) + b*e^(-m*x)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.190262, size = 26, normalized size = 0.84 \[ \frac{\operatorname{RootSum}{\left (4 z^{2} a b + 1, \left ( i \mapsto i \log{\left (- 2 i a + e^{- m x} \right )} \right )\right )}}{m} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b/exp(m*x)+a*exp(m*x)),x)
[Out]
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GIAC/XCAS [A] time = 0.20011, size = 28, normalized size = 0.9 \[ \frac{\arctan \left (\frac{a e^{\left (m x\right )}}{\sqrt{a b}}\right )}{\sqrt{a b} m} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a*e^(m*x) + b*e^(-m*x)),x, algorithm="giac")
[Out]