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

Optimal. Leaf size=18 \[ \frac{\tan ^{-1}\left (\sqrt{\frac{3}{2}} e^x\right )}{\sqrt{6}} \]

[Out]

ArcTan[Sqrt[3/2]*E^x]/Sqrt[6]

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Rubi [A]  time = 0.0357168, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{\tan ^{-1}\left (\sqrt{\frac{3}{2}} e^x\right )}{\sqrt{6}} \]

Antiderivative was successfully verified.

[In]  Int[E^x/(2 + 3*E^(2*x)),x]

[Out]

ArcTan[Sqrt[3/2]*E^x]/Sqrt[6]

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Rubi in Sympy [A]  time = 3.15427, size = 17, normalized size = 0.94 \[ \frac{\sqrt{6} \operatorname{atan}{\left (\frac{\sqrt{6} e^{x}}{2} \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(6)*atan(sqrt(6)*exp(x)/2)/6

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Mathematica [A]  time = 0.0131084, size = 18, normalized size = 1. \[ \frac{\tan ^{-1}\left (\sqrt{\frac{3}{2}} e^x\right )}{\sqrt{6}} \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[Sqrt[3/2]*E^x]/Sqrt[6]

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Maple [A]  time = 0.005, size = 14, normalized size = 0.8 \[{\frac{\sqrt{6}}{6}\arctan \left ({\frac{{{\rm e}^{x}}\sqrt{6}}{2}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(x)/(2+3*exp(2*x)),x)

[Out]

1/6*arctan(1/2*exp(x)*6^(1/2))*6^(1/2)

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Maxima [A]  time = 1.54985, size = 18, normalized size = 1. \[ \frac{1}{6} \, \sqrt{6} \arctan \left (\frac{1}{2} \, \sqrt{6} e^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*sqrt(6)*arctan(1/2*sqrt(6)*e^x)

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Fricas [A]  time = 0.207156, size = 18, normalized size = 1. \[ \frac{1}{6} \, \sqrt{6} \arctan \left (\frac{1}{2} \, \sqrt{6} e^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*sqrt(6)*arctan(1/2*sqrt(6)*e^x)

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Sympy [A]  time = 0.113463, size = 15, normalized size = 0.83 \[ \operatorname{RootSum}{\left (24 z^{2} + 1, \left ( i \mapsto i \log{\left (4 i + e^{x} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(24*_z**2 + 1, Lambda(_i, _i*log(4*_i + exp(x))))

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GIAC/XCAS [A]  time = 0.207173, size = 18, normalized size = 1. \[ \frac{1}{6} \, \sqrt{6} \arctan \left (\frac{1}{2} \, \sqrt{6} e^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*sqrt(6)*arctan(1/2*sqrt(6)*e^x)