Optimal. Leaf size=17 \[ x+\frac{1}{e^x+1}-\log \left (e^x+1\right ) \]
[Out]
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Rubi [A] time = 0.0294842, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ x+\frac{1}{e^x+1}-\log \left (e^x+1\right ) \]
Antiderivative was successfully verified.
[In] Int[(1 + 2*E^x + E^(2*x))^(-1),x]
[Out]
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Rubi in Sympy [A] time = 7.38376, size = 17, normalized size = 1. \[ - \log{\left (e^{x} + 1 \right )} + \log{\left (e^{x} \right )} + \frac{1}{e^{x} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(1+2*exp(x)+exp(2*x)),x)
[Out]
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Mathematica [A] time = 0.0167562, size = 17, normalized size = 1. \[ x+\frac{1}{e^x+1}-\log \left (e^x+1\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(1 + 2*E^x + E^(2*x))^(-1),x]
[Out]
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Maple [A] time = 0.013, size = 18, normalized size = 1.1 \[ \left ( 1+{{\rm e}^{x}} \right ) ^{-1}-\ln \left ( 1+{{\rm e}^{x}} \right ) +\ln \left ({{\rm e}^{x}} \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(1+2*exp(x)+exp(2*x)),x)
[Out]
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Maxima [A] time = 0.780711, size = 20, normalized size = 1.18 \[ x + \frac{1}{e^{x} + 1} - \log \left (e^{x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(e^(2*x) + 2*e^x + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.246991, size = 34, normalized size = 2. \[ \frac{x e^{x} -{\left (e^{x} + 1\right )} \log \left (e^{x} + 1\right ) + x + 1}{e^{x} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(e^(2*x) + 2*e^x + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.065228, size = 14, normalized size = 0.82 \[ x - \log{\left (e^{x} + 1 \right )} + \frac{1}{e^{x} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(1+2*exp(x)+exp(2*x)),x)
[Out]
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GIAC/XCAS [A] time = 0.237677, size = 20, normalized size = 1.18 \[ x + \frac{1}{e^{x} + 1} -{\rm ln}\left (e^{x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(e^(2*x) + 2*e^x + 1),x, algorithm="giac")
[Out]