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

Optimal. Leaf size=22 \[ \frac{e^x}{b}-\frac{a \log \left (a+b e^x\right )}{b^2} \]

[Out]

E^x/b - (a*Log[a + b*E^x])/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]

E^x/b - (a*Log[a + b*E^x])/b^2

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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)

[Out]

-a*log(a + b*exp(x))/b**2 + Integral(1/b, (x, exp(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]

E^x/b - (a*Log[a + b*E^x])/b^2

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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]

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

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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]

e^x/b - a*log(b*e^x + a)/b^2

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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]

(b*e^x - a*log(b*e^x + a))/b^2

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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]

-a*log(a/b + exp(x))/b**2 + Piecewise((exp(x)/b, Ne(b, 0)), (x/b, True))

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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]

e^x/b - a*ln(abs(b*e^x + a))/b^2