3.19 \(\int \frac{e^{2 x}}{\left (a+b e^x\right )^2} \, dx\)

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]

a/(b^2*(a + b*E^x)) + Log[a + b*E^x]/b^2

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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]

a/(b^2*(a + b*E^x)) + Log[a + b*E^x]/b^2

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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]

a/(b**2*(a + b*exp(x))) + log(a + b*exp(x))/b**2

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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]

(a/(a + b*E^x) + Log[a + b*E^x])/b^2

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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]

a/b^2/(a+b*exp(x))+ln(a+b*exp(x))/b^2

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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]

a/(b^3*e^x + a*b^2) + log(b*e^x + a)/b^2

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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]

((b*e^x + a)*log(b*e^x + a) + a)/(b^3*e^x + a*b^2)

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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]

a/(a*b**2 + b**3*exp(x)) + log(a/b + exp(x))/b**2

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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]

ln(abs(b*e^x + a))/b^2 + a/((b*e^x + a)*b^2)