3.689 \(\int \frac{e^x}{-1-8 e^x+e^{2 x}} \, dx\)

Optimal. Leaf size=20 \[ \frac{\tanh ^{-1}\left (\frac{4-e^x}{\sqrt{17}}\right )}{\sqrt{17}} \]

[Out]

ArcTanh[(4 - E^x)/Sqrt[17]]/Sqrt[17]

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Rubi [A]  time = 0.0658125, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{\tanh ^{-1}\left (\frac{4-e^x}{\sqrt{17}}\right )}{\sqrt{17}} \]

Antiderivative was successfully verified.

[In]  Int[E^x/(-1 - 8*E^x + E^(2*x)),x]

[Out]

ArcTanh[(4 - E^x)/Sqrt[17]]/Sqrt[17]

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Rubi in Sympy [A]  time = 15.9802, size = 22, normalized size = 1.1 \[ - \frac{\sqrt{17} \operatorname{atanh}{\left (\sqrt{17} \left (\frac{e^{x}}{17} - \frac{4}{17}\right ) \right )}}{17} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(x)/(-1-8*exp(x)+exp(2*x)),x)

[Out]

-sqrt(17)*atanh(sqrt(17)*(exp(x)/17 - 4/17))/17

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Mathematica [A]  time = 0.0291338, size = 36, normalized size = 1.8 \[ \frac{\log \left (-e^x+4+\sqrt{17}\right )-\log \left (e^x-4+\sqrt{17}\right )}{2 \sqrt{17}} \]

Antiderivative was successfully verified.

[In]  Integrate[E^x/(-1 - 8*E^x + E^(2*x)),x]

[Out]

(Log[4 + Sqrt[17] - E^x] - Log[-4 + Sqrt[17] + E^x])/(2*Sqrt[17])

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Maple [A]  time = 0.005, size = 18, normalized size = 0.9 \[ -{\frac{\sqrt{17}}{17}{\it Artanh} \left ({\frac{ \left ( 2\,{{\rm e}^{x}}-8 \right ) \sqrt{17}}{34}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(x)/(-1-8*exp(x)+exp(2*x)),x)

[Out]

-1/17*17^(1/2)*arctanh(1/34*(2*exp(x)-8)*17^(1/2))

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Maxima [A]  time = 0.885497, size = 35, normalized size = 1.75 \[ \frac{1}{34} \, \sqrt{17} \log \left (-\frac{\sqrt{17} - e^{x} + 4}{\sqrt{17} + e^{x} - 4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^x/(e^(2*x) - 8*e^x - 1),x, algorithm="maxima")

[Out]

1/34*sqrt(17)*log(-(sqrt(17) - e^x + 4)/(sqrt(17) + e^x - 4))

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Fricas [A]  time = 0.243382, size = 63, normalized size = 3.15 \[ \frac{1}{34} \, \sqrt{17} \log \left (-\frac{2 \,{\left (4 \, \sqrt{17} + 17\right )} e^{x} - \sqrt{17} e^{\left (2 \, x\right )} - 33 \, \sqrt{17} - 136}{e^{\left (2 \, x\right )} - 8 \, e^{x} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^x/(e^(2*x) - 8*e^x - 1),x, algorithm="fricas")

[Out]

1/34*sqrt(17)*log(-(2*(4*sqrt(17) + 17)*e^x - sqrt(17)*e^(2*x) - 33*sqrt(17) - 1
36)/(e^(2*x) - 8*e^x - 1))

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Sympy [A]  time = 0.124389, size = 17, normalized size = 0.85 \[ \operatorname{RootSum}{\left (68 z^{2} - 1, \left ( i \mapsto i \log{\left (- 34 i + e^{x} - 4 \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(x)/(-1-8*exp(x)+exp(2*x)),x)

[Out]

RootSum(68*_z**2 - 1, Lambda(_i, _i*log(-34*_i + exp(x) - 4)))

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GIAC/XCAS [A]  time = 0.320522, size = 45, normalized size = 2.25 \[ \frac{1}{34} \, \sqrt{17}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{17} + 2 \, e^{x} - 8 \right |}}{{\left | 2 \, \sqrt{17} + 2 \, e^{x} - 8 \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^x/(e^(2*x) - 8*e^x - 1),x, algorithm="giac")

[Out]

1/34*sqrt(17)*ln(abs(-2*sqrt(17) + 2*e^x - 8)/abs(2*sqrt(17) + 2*e^x - 8))