Optimal. Leaf size=36 \[ -\frac{2 \tanh ^{-1}\left (\frac{a+2 c e^x}{\sqrt{a^2-4 b c}}\right )}{\sqrt{a^2-4 b c}} \]
[Out]
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Rubi [A] time = 0.109068, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{2 \tanh ^{-1}\left (\frac{a+2 c e^x}{\sqrt{a^2-4 b c}}\right )}{\sqrt{a^2-4 b c}} \]
Antiderivative was successfully verified.
[In] Int[(a + b/E^x + c*E^x)^(-1),x]
[Out]
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Rubi in Sympy [A] time = 9.86818, size = 36, normalized size = 1. \[ - \frac{2 \operatorname{atanh}{\left (\frac{a + 2 c e^{x}}{\sqrt{a^{2} - 4 b c}} \right )}}{\sqrt{a^{2} - 4 b c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b/exp(x)+c*exp(x)),x)
[Out]
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Mathematica [A] time = 0.0347236, size = 40, normalized size = 1.11 \[ \frac{2 \tan ^{-1}\left (\frac{a+2 c e^x}{\sqrt{4 b c-a^2}}\right )}{\sqrt{4 b c-a^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b/E^x + c*E^x)^(-1),x]
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Maple [A] time = 0.007, size = 36, normalized size = 1. \[ 2\,{\frac{1}{\sqrt{-{a}^{2}+4\,cb}}\arctan \left ({\frac{a+2\,c{{\rm e}^{x}}}{\sqrt{-{a}^{2}+4\,cb}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b/exp(x)+c*exp(x)),x)
[Out]
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*e^(-x) + c*e^x + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.254213, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (-\frac{a^{3} - 4 \, a b c + 2 \,{\left (a^{2} c - 4 \, b c^{2}\right )} e^{x} -{\left (2 \, c^{2} e^{\left (2 \, x\right )} + 2 \, a c e^{x} + a^{2} - 2 \, b c\right )} \sqrt{a^{2} - 4 \, b c}}{c e^{\left (2 \, x\right )} + a e^{x} + b}\right )}{\sqrt{a^{2} - 4 \, b c}}, \frac{2 \, \arctan \left (-\frac{\sqrt{-a^{2} + 4 \, b c}{\left (2 \, c e^{x} + a\right )}}{a^{2} - 4 \, b c}\right )}{\sqrt{-a^{2} + 4 \, b c}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*e^(-x) + c*e^x + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.317193, size = 36, normalized size = 1. \[ \operatorname{RootSum}{\left (z^{2} \left (a^{2} - 4 b c\right ) - 1, \left ( i \mapsto i \log{\left (e^{x} + \frac{- i a^{2} + 4 i b c + a}{2 c} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b/exp(x)+c*exp(x)),x)
[Out]
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GIAC/XCAS [A] time = 0.230236, size = 47, normalized size = 1.31 \[ \frac{2 \, \arctan \left (\frac{2 \, c e^{x} + a}{\sqrt{-a^{2} + 4 \, b c}}\right )}{\sqrt{-a^{2} + 4 \, b c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*e^(-x) + c*e^x + a),x, algorithm="giac")
[Out]