3.507 \(\int \frac{1}{-1+e^x+e^{2 x}} \, dx\)

Optimal. Leaf size=56 \[ -x+\frac{1}{10} \left (5+\sqrt{5}\right ) \log \left (2 e^x+1-\sqrt{5}\right )+\frac{1}{10} \left (5-\sqrt{5}\right ) \log \left (2 e^x+1+\sqrt{5}\right ) \]

[Out]

-x + ((5 + Sqrt[5])*Log[1 - Sqrt[5] + 2*E^x])/10 + ((5 - Sqrt[5])*Log[1 + Sqrt[5
] + 2*E^x])/10

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Rubi [A]  time = 0.0681013, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417 \[ -x+\frac{1}{10} \left (5+\sqrt{5}\right ) \log \left (2 e^x+1-\sqrt{5}\right )+\frac{1}{10} \left (5-\sqrt{5}\right ) \log \left (2 e^x+1+\sqrt{5}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(-1 + E^x + E^(2*x))^(-1),x]

[Out]

-x + ((5 + Sqrt[5])*Log[1 - Sqrt[5] + 2*E^x])/10 + ((5 - Sqrt[5])*Log[1 + Sqrt[5
] + 2*E^x])/10

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Rubi in Sympy [A]  time = 8.97086, size = 65, normalized size = 1.16 \[ - \frac{\sqrt{5} \left (- \frac{\sqrt{5}}{2} + \frac{1}{2}\right ) \log{\left (2 e^{x} + 1 + \sqrt{5} \right )}}{5} + \frac{\sqrt{5} \left (\frac{1}{2} + \frac{\sqrt{5}}{2}\right ) \log{\left (2 e^{x} - \sqrt{5} + 1 \right )}}{5} - \log{\left (e^{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-1+exp(x)+exp(2*x)),x)

[Out]

-sqrt(5)*(-sqrt(5)/2 + 1/2)*log(2*exp(x) + 1 + sqrt(5))/5 + sqrt(5)*(1/2 + sqrt(
5)/2)*log(2*exp(x) - sqrt(5) + 1)/5 - log(exp(x))

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Mathematica [A]  time = 0.0719786, size = 44, normalized size = 0.79 \[ -x+\frac{1}{2} \log \left (-e^x-e^{2 x}+1\right )-\frac{\tanh ^{-1}\left (\frac{2 e^x+1}{\sqrt{5}}\right )}{\sqrt{5}} \]

Antiderivative was successfully verified.

[In]  Integrate[(-1 + E^x + E^(2*x))^(-1),x]

[Out]

-x - ArcTanh[(1 + 2*E^x)/Sqrt[5]]/Sqrt[5] + Log[1 - E^x - E^(2*x)]/2

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Maple [A]  time = 0.009, size = 35, normalized size = 0.6 \[{\frac{\ln \left ( -1+{{\rm e}^{x}}+ \left ({{\rm e}^{x}} \right ) ^{2} \right ) }{2}}-{\frac{\sqrt{5}}{5}{\it Artanh} \left ({\frac{ \left ( 1+2\,{{\rm e}^{x}} \right ) \sqrt{5}}{5}} \right ) }-\ln \left ({{\rm e}^{x}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-1+exp(x)+exp(2*x)),x)

[Out]

1/2*ln(-1+exp(x)+exp(x)^2)-1/5*5^(1/2)*arctanh(1/5*(1+2*exp(x))*5^(1/2))-ln(exp(
x))

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Maxima [A]  time = 0.849671, size = 58, normalized size = 1.04 \[ \frac{1}{10} \, \sqrt{5} \log \left (-\frac{\sqrt{5} - 2 \, e^{x} - 1}{\sqrt{5} + 2 \, e^{x} + 1}\right ) - x + \frac{1}{2} \, \log \left (e^{\left (2 \, x\right )} + e^{x} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e^(2*x) + e^x - 1),x, algorithm="maxima")

[Out]

1/10*sqrt(5)*log(-(sqrt(5) - 2*e^x - 1)/(sqrt(5) + 2*e^x + 1)) - x + 1/2*log(e^(
2*x) + e^x - 1)

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Fricas [A]  time = 0.274001, size = 88, normalized size = 1.57 \[ -\frac{1}{10} \, \sqrt{5}{\left (2 \, \sqrt{5} x - \sqrt{5} \log \left (e^{\left (2 \, x\right )} + e^{x} - 1\right ) - \log \left (\frac{2 \,{\left (\sqrt{5} - 5\right )} e^{x} + 2 \, \sqrt{5} e^{\left (2 \, x\right )} + 3 \, \sqrt{5} - 5}{e^{\left (2 \, x\right )} + e^{x} - 1}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e^(2*x) + e^x - 1),x, algorithm="fricas")

[Out]

-1/10*sqrt(5)*(2*sqrt(5)*x - sqrt(5)*log(e^(2*x) + e^x - 1) - log((2*(sqrt(5) -
5)*e^x + 2*sqrt(5)*e^(2*x) + 3*sqrt(5) - 5)/(e^(2*x) + e^x - 1)))

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Sympy [A]  time = 0.129005, size = 22, normalized size = 0.39 \[ - x + \operatorname{RootSum}{\left (5 z^{2} - 5 z + 1, \left ( i \mapsto i \log{\left (- 5 i + e^{x} + 3 \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-1+exp(x)+exp(2*x)),x)

[Out]

-x + RootSum(5*_z**2 - 5*_z + 1, Lambda(_i, _i*log(-5*_i + exp(x) + 3)))

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GIAC/XCAS [A]  time = 0.256165, size = 62, normalized size = 1.11 \[ \frac{1}{10} \, \sqrt{5}{\rm ln}\left (\frac{{\left | -\sqrt{5} + 2 \, e^{x} + 1 \right |}}{\sqrt{5} + 2 \, e^{x} + 1}\right ) - x + \frac{1}{2} \,{\rm ln}\left ({\left | e^{\left (2 \, x\right )} + e^{x} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e^(2*x) + e^x - 1),x, algorithm="giac")

[Out]

1/10*sqrt(5)*ln(abs(-sqrt(5) + 2*e^x + 1)/(sqrt(5) + 2*e^x + 1)) - x + 1/2*ln(ab
s(e^(2*x) + e^x - 1))