3.620 \(\int \frac{e^x}{4+e^{2 x}} \, dx\)

Optimal. Leaf size=12 \[ \frac{1}{2} \tan ^{-1}\left (\frac{e^x}{2}\right ) \]

[Out]

ArcTan[E^x/2]/2

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Rubi [A]  time = 0.0308902, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{1}{2} \tan ^{-1}\left (\frac{e^x}{2}\right ) \]

Antiderivative was successfully verified.

[In]  Int[E^x/(4 + E^(2*x)),x]

[Out]

ArcTan[E^x/2]/2

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Rubi in Sympy [A]  time = 5.64931, size = 7, normalized size = 0.58 \[ \frac{\operatorname{atan}{\left (\frac{e^{x}}{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(x)/(4+exp(2*x)),x)

[Out]

atan(exp(x)/2)/2

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Mathematica [A]  time = 0.00615455, size = 12, normalized size = 1. \[ -\frac{1}{2} \tan ^{-1}\left (2 e^{-x}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[E^x/(4 + E^(2*x)),x]

[Out]

-ArcTan[2/E^x]/2

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Maple [A]  time = 0.004, size = 8, normalized size = 0.7 \[{\frac{1}{2}\arctan \left ({\frac{{{\rm e}^{x}}}{2}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(x)/(4+exp(2*x)),x)

[Out]

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

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Maxima [A]  time = 0.855386, size = 9, normalized size = 0.75 \[ \frac{1}{2} \, \arctan \left (\frac{1}{2} \, e^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^x/(e^(2*x) + 4),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.255744, size = 9, normalized size = 0.75 \[ \frac{1}{2} \, \arctan \left (\frac{1}{2} \, e^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^x/(e^(2*x) + 4),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.100897, size = 15, normalized size = 1.25 \[ \operatorname{RootSum}{\left (16 z^{2} + 1, \left ( i \mapsto i \log{\left (8 i + e^{x} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(x)/(4+exp(2*x)),x)

[Out]

RootSum(16*_z**2 + 1, Lambda(_i, _i*log(8*_i + exp(x))))

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GIAC/XCAS [A]  time = 0.227604, size = 9, normalized size = 0.75 \[ \frac{1}{2} \, \arctan \left (\frac{1}{2} \, e^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^x/(e^(2*x) + 4),x, algorithm="giac")

[Out]

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