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

Optimal. Leaf size=44 \[ \frac{x}{3}-\frac{1}{6} \log \left (3 e^x+e^{2 x}+3\right )-\frac{\tan ^{-1}\left (\frac{2 e^x+3}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

x/3 - ArcTan[(3 + 2*E^x)/Sqrt[3]]/Sqrt[3] - Log[3 + 3*E^x + E^(2*x)]/6

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Rubi [A]  time = 0.0706599, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ \frac{x}{3}-\frac{1}{6} \log \left (3 e^x+e^{2 x}+3\right )-\frac{\tan ^{-1}\left (\frac{2 e^x+3}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

x/3 - ArcTan[(3 + 2*E^x)/Sqrt[3]]/Sqrt[3] - Log[3 + 3*E^x + E^(2*x)]/6

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Rubi in Sympy [A]  time = 10.4803, size = 42, normalized size = 0.95 \[ - \frac{\log{\left (e^{2 x} + 3 e^{x} + 3 \right )}}{6} + \frac{\log{\left (e^{x} \right )}}{3} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 e^{x}}{3} + 1\right ) \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(exp(2*x) + 3*exp(x) + 3)/6 + log(exp(x))/3 - sqrt(3)*atan(sqrt(3)*(2*exp(x)
/3 + 1))/3

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Mathematica [A]  time = 0.0520024, size = 44, normalized size = 1. \[ \frac{x}{3}-\frac{1}{6} \log \left (3 e^x+e^{2 x}+3\right )-\frac{\tan ^{-1}\left (\frac{2 e^x+3}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

x/3 - ArcTan[(3 + 2*E^x)/Sqrt[3]]/Sqrt[3] - Log[3 + 3*E^x + E^(2*x)]/6

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Maple [A]  time = 0.009, size = 37, normalized size = 0.8 \[{\frac{\ln \left ({{\rm e}^{x}} \right ) }{3}}-{\frac{\ln \left ( 3+3\,{{\rm e}^{x}}+ \left ({{\rm e}^{x}} \right ) ^{2} \right ) }{6}}-{\frac{\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 3+2\,{{\rm e}^{x}} \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(3+3*exp(x)+exp(2*x)),x)

[Out]

1/3*ln(exp(x))-1/6*ln(3+3*exp(x)+exp(x)^2)-1/3*arctan(1/3*(3+2*exp(x))*3^(1/2))*
3^(1/2)

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Maxima [A]  time = 0.856124, size = 46, normalized size = 1.05 \[ -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, e^{x} + 3\right )}\right ) + \frac{1}{3} \, x - \frac{1}{6} \, \log \left (e^{\left (2 \, x\right )} + 3 \, e^{x} + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*e^x + 3)) + 1/3*x - 1/6*log(e^(2*x) + 3*e^x +
3)

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Fricas [A]  time = 0.293075, size = 57, normalized size = 1.3 \[ \frac{1}{18} \, \sqrt{3}{\left (2 \, \sqrt{3} x - \sqrt{3} \log \left (e^{\left (2 \, x\right )} + 3 \, e^{x} + 3\right ) - 6 \, \arctan \left (\frac{2}{3} \, \sqrt{3} e^{x} + \sqrt{3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/18*sqrt(3)*(2*sqrt(3)*x - sqrt(3)*log(e^(2*x) + 3*e^x + 3) - 6*arctan(2/3*sqrt
(3)*e^x + sqrt(3)))

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Sympy [A]  time = 0.128036, size = 24, normalized size = 0.55 \[ \frac{x}{3} + \operatorname{RootSum}{\left (9 z^{2} + 3 z + 1, \left ( i \mapsto i \log{\left (- 3 i + e^{x} + 1 \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/3 + RootSum(9*_z**2 + 3*_z + 1, Lambda(_i, _i*log(-3*_i + exp(x) + 1)))

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GIAC/XCAS [A]  time = 0.251116, size = 46, normalized size = 1.05 \[ -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, e^{x} + 3\right )}\right ) + \frac{1}{3} \, x - \frac{1}{6} \,{\rm ln}\left (e^{\left (2 \, x\right )} + 3 \, e^{x} + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*e^x + 3)) + 1/3*x - 1/6*ln(e^(2*x) + 3*e^x +
3)