Optimal. Leaf size=27 \[ \frac{a}{b^2 \left (a+b e^x\right )}+\frac{\log \left (a+b e^x\right )}{b^2} \]
[Out]
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Rubi [A] time = 0.0604489, antiderivative size = 27, 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 \[ \frac{a}{b^2 \left (a+b e^x\right )}+\frac{\log \left (a+b e^x\right )}{b^2} \]
Antiderivative was successfully verified.
[In] Int[E^(2*x)/(a + b*E^x)^2,x]
[Out]
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Rubi in Sympy [A] time = 10.0582, size = 22, normalized size = 0.81 \[ \frac{a}{b^{2} \left (a + b e^{x}\right )} + \frac{\log{\left (a + b e^{x} \right )}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(exp(2*x)/(a+b*exp(x))**2,x)
[Out]
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Mathematica [A] time = 0.0230116, size = 24, normalized size = 0.89 \[ \frac{\frac{a}{a+b e^x}+\log \left (a+b e^x\right )}{b^2} \]
Antiderivative was successfully verified.
[In] Integrate[E^(2*x)/(a + b*E^x)^2,x]
[Out]
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Maple [A] time = 0.01, size = 26, normalized size = 1. \[{\frac{a}{{b}^{2} \left ( a+b{{\rm e}^{x}} \right ) }}+{\frac{\ln \left ( a+b{{\rm e}^{x}} \right ) }{{b}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(exp(2*x)/(a+b*exp(x))^2,x)
[Out]
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Maxima [A] time = 0.777959, size = 38, normalized size = 1.41 \[ \frac{a}{b^{3} e^{x} + a b^{2}} + \frac{\log \left (b e^{x} + a\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(e^(2*x)/(b*e^x + a)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.257842, size = 42, normalized size = 1.56 \[ \frac{{\left (b e^{x} + a\right )} \log \left (b e^{x} + a\right ) + a}{b^{3} e^{x} + a b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(e^(2*x)/(b*e^x + a)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.273961, size = 24, normalized size = 0.89 \[ \frac{a}{a b^{2} + b^{3} e^{x}} + \frac{\log{\left (\frac{a}{b} + e^{x} \right )}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(exp(2*x)/(a+b*exp(x))**2,x)
[Out]
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GIAC/XCAS [A] time = 0.254986, size = 35, normalized size = 1.3 \[ \frac{{\rm ln}\left ({\left | b e^{x} + a \right |}\right )}{b^{2}} + \frac{a}{{\left (b e^{x} + a\right )} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(e^(2*x)/(b*e^x + a)^2,x, algorithm="giac")
[Out]