Optimal. Leaf size=22 \[ \frac{e^x}{b}-\frac{a \log \left (a+b e^x\right )}{b^2} \]
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Rubi [A] time = 0.054025, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{e^x}{b}-\frac{a \log \left (a+b e^x\right )}{b^2} \]
Antiderivative was successfully verified.
[In] Int[E^(2*x)/(a + b*E^x),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{a \log{\left (a + b e^{x} \right )}}{b^{2}} + \int ^{e^{x}} \frac{1}{b}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(exp(2*x)/(a+b*exp(x)),x)
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Mathematica [A] time = 0.0113405, size = 22, normalized size = 1. \[ \frac{e^x}{b}-\frac{a \log \left (a+b e^x\right )}{b^2} \]
Antiderivative was successfully verified.
[In] Integrate[E^(2*x)/(a + b*E^x),x]
[Out]
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Maple [A] time = 0.006, size = 21, normalized size = 1. \[{\frac{{{\rm e}^{x}}}{b}}-{\frac{a\ln \left ( a+b{{\rm e}^{x}} \right ) }{{b}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(exp(2*x)/(a+b*exp(x)),x)
[Out]
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Maxima [A] time = 0.766471, size = 27, normalized size = 1.23 \[ \frac{e^{x}}{b} - \frac{a \log \left (b e^{x} + a\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(e^(2*x)/(b*e^x + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.245301, size = 26, normalized size = 1.18 \[ \frac{b e^{x} - a \log \left (b e^{x} + a\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(e^(2*x)/(b*e^x + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.258955, size = 20, normalized size = 0.91 \[ - \frac{a \log{\left (\frac{a}{b} + e^{x} \right )}}{b^{2}} + \begin{cases} \frac{e^{x}}{b} & \text{for}\: b \neq 0 \\\frac{x}{b} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(exp(2*x)/(a+b*exp(x)),x)
[Out]
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GIAC/XCAS [A] time = 0.230413, size = 28, normalized size = 1.27 \[ \frac{e^{x}}{b} - \frac{a{\rm ln}\left ({\left | b e^{x} + a \right |}\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(e^(2*x)/(b*e^x + a),x, algorithm="giac")
[Out]