3.526 \(\int \frac{e^x+e^{5 x}}{-1+e^x-e^{2 x}+e^{3 x}} \, dx\)

Optimal. Leaf size=39 \[ e^x+\frac{e^{2 x}}{2}+\log \left (1-e^x\right )-\frac{1}{2} \log \left (e^{2 x}+1\right )-\tan ^{-1}\left (e^x\right ) \]

[Out]

E^x + E^(2*x)/2 - ArcTan[E^x] + Log[1 - E^x] - Log[1 + E^(2*x)]/2

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Rubi [A]  time = 0.104901, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172 \[ e^x+\frac{e^{2 x}}{2}+\log \left (1-e^x\right )-\frac{1}{2} \log \left (e^{2 x}+1\right )-\tan ^{-1}\left (e^x\right ) \]

Antiderivative was successfully verified.

[In]  Int[(E^x + E^(5*x))/(-1 + E^x - E^(2*x) + E^(3*x)),x]

[Out]

E^x + E^(2*x)/2 - ArcTan[E^x] + Log[1 - E^x] - Log[1 + E^(2*x)]/2

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{e^{5 x} + e^{x}}{e^{3 x} - e^{2 x} + e^{x} - 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((exp(x)+exp(5*x))/(-1+exp(x)-exp(2*x)+exp(3*x)),x)

[Out]

Integral((exp(5*x) + exp(x))/(exp(3*x) - exp(2*x) + exp(x) - 1), x)

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Mathematica [A]  time = 0.0322495, size = 39, normalized size = 1. \[ e^x+\frac{e^{2 x}}{2}+\log \left (1-e^x\right )-\frac{1}{2} \log \left (e^{2 x}+1\right )-\tan ^{-1}\left (e^x\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(E^x + E^(5*x))/(-1 + E^x - E^(2*x) + E^(3*x)),x]

[Out]

E^x + E^(2*x)/2 - ArcTan[E^x] + Log[1 - E^x] - Log[1 + E^(2*x)]/2

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Maple [A]  time = 0.019, size = 29, normalized size = 0.7 \[ -{\frac{\ln \left ( \left ({{\rm e}^{x}} \right ) ^{2}+1 \right ) }{2}}-\arctan \left ({{\rm e}^{x}} \right ) +\ln \left ( -1+{{\rm e}^{x}} \right ) +{{\rm e}^{x}}+{\frac{ \left ({{\rm e}^{x}} \right ) ^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((exp(x)+exp(5*x))/(-1+exp(x)-exp(2*x)+exp(3*x)),x)

[Out]

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

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Maxima [A]  time = 1.57276, size = 38, normalized size = 0.97 \[ -\arctan \left (e^{x}\right ) + \frac{1}{2} \, e^{\left (2 \, x\right )} + e^{x} - \frac{1}{2} \, \log \left (e^{\left (2 \, x\right )} + 1\right ) + \log \left (e^{x} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e^(5*x) + e^x)/(e^(3*x) - e^(2*x) + e^x - 1),x, algorithm="maxima")

[Out]

-arctan(e^x) + 1/2*e^(2*x) + e^x - 1/2*log(e^(2*x) + 1) + log(e^x - 1)

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Fricas [A]  time = 0.230679, size = 38, normalized size = 0.97 \[ -\arctan \left (e^{x}\right ) + \frac{1}{2} \, e^{\left (2 \, x\right )} + e^{x} - \frac{1}{2} \, \log \left (e^{\left (2 \, x\right )} + 1\right ) + \log \left (e^{x} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e^(5*x) + e^x)/(e^(3*x) - e^(2*x) + e^x - 1),x, algorithm="fricas")

[Out]

-arctan(e^x) + 1/2*e^(2*x) + e^x - 1/2*log(e^(2*x) + 1) + log(e^x - 1)

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Sympy [A]  time = 0.240272, size = 48, normalized size = 1.23 \[ \frac{e^{2 x}}{2} + e^{x} + \log{\left (e^{x} - 1 \right )} + \operatorname{RootSum}{\left (2 z^{2} + 2 z + 1, \left ( i \mapsto i \log{\left (\frac{4 i^{2}}{5} - \frac{6 i}{5} + e^{x} - \frac{3}{5} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((exp(x)+exp(5*x))/(-1+exp(x)-exp(2*x)+exp(3*x)),x)

[Out]

exp(2*x)/2 + exp(x) + log(exp(x) - 1) + RootSum(2*_z**2 + 2*_z + 1, Lambda(_i, _
i*log(4*_i**2/5 - 6*_i/5 + exp(x) - 3/5)))

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GIAC/XCAS [A]  time = 0.199046, size = 39, normalized size = 1. \[ -\arctan \left (e^{x}\right ) + \frac{1}{2} \, e^{\left (2 \, x\right )} + e^{x} - \frac{1}{2} \,{\rm ln}\left (e^{\left (2 \, x\right )} + 1\right ) +{\rm ln}\left ({\left | e^{x} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e^(5*x) + e^x)/(e^(3*x) - e^(2*x) + e^x - 1),x, algorithm="giac")

[Out]

-arctan(e^x) + 1/2*e^(2*x) + e^x - 1/2*ln(e^(2*x) + 1) + ln(abs(e^x - 1))