Optimal. Leaf size=21 \[ \frac{e^{2 x}}{2 a \left (a+b e^x\right )^2} \]
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Rubi [A] time = 0.0408097, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{e^{2 x}}{2 a \left (a+b e^x\right )^2} \]
Antiderivative was successfully verified.
[In] Int[E^(2*x)/(a + b*E^x)^3,x]
[Out]
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Rubi in Sympy [A] time = 6.89898, size = 15, normalized size = 0.71 \[ \frac{e^{2 x}}{2 a \left (a + b e^{x}\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(exp(2*x)/(a+b*exp(x))**3,x)
[Out]
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Mathematica [A] time = 0.0156651, size = 24, normalized size = 1.14 \[ -\frac{a+2 b e^x}{2 b^2 \left (a+b e^x\right )^2} \]
Antiderivative was successfully verified.
[In] Integrate[E^(2*x)/(a + b*E^x)^3,x]
[Out]
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Maple [A] time = 0.01, size = 29, normalized size = 1.4 \[{\frac{a}{2\,{b}^{2} \left ( a+b{{\rm e}^{x}} \right ) ^{2}}}-{\frac{1}{{b}^{2} \left ( a+b{{\rm e}^{x}} \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(exp(2*x)/(a+b*exp(x))^3,x)
[Out]
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Maxima [A] time = 0.765769, size = 82, normalized size = 3.9 \[ -\frac{b e^{x}}{b^{4} e^{\left (2 \, x\right )} + 2 \, a b^{3} e^{x} + a^{2} b^{2}} - \frac{a}{2 \,{\left (b^{4} e^{\left (2 \, x\right )} + 2 \, a b^{3} e^{x} + a^{2} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(e^(2*x)/(b*e^x + a)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.265172, size = 47, normalized size = 2.24 \[ -\frac{2 \, b e^{x} + a}{2 \,{\left (b^{4} e^{\left (2 \, x\right )} + 2 \, a b^{3} e^{x} + a^{2} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(e^(2*x)/(b*e^x + a)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.249224, size = 37, normalized size = 1.76 \[ \frac{- a - 2 b e^{x}}{2 a^{2} b^{2} + 4 a b^{3} e^{x} + 2 b^{4} e^{2 x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(exp(2*x)/(a+b*exp(x))**3,x)
[Out]
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GIAC/XCAS [A] time = 0.323473, size = 27, normalized size = 1.29 \[ -\frac{2 \, b e^{x} + a}{2 \,{\left (b e^{x} + a\right )}^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(e^(2*x)/(b*e^x + a)^3,x, algorithm="giac")
[Out]