3.632 \(\int \frac{e^{2 x}}{1+e^{4 x}} \, dx\)

Optimal. Leaf size=10 \[ \frac{1}{2} \tan ^{-1}\left (e^{2 x}\right ) \]

[Out]

ArcTan[E^(2*x)]/2

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Rubi [A]  time = 0.033772, antiderivative size = 10, 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 \[ \frac{1}{2} \tan ^{-1}\left (e^{2 x}\right ) \]

Antiderivative was successfully verified.

[In]  Int[E^(2*x)/(1 + E^(4*x)),x]

[Out]

ArcTan[E^(2*x)]/2

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Rubi in Sympy [A]  time = 5.8892, size = 7, normalized size = 0.7 \[ \frac{\operatorname{atan}{\left (e^{2 x} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(2*x)/(1+exp(4*x)),x)

[Out]

atan(exp(2*x))/2

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Mathematica [A]  time = 0.00583009, size = 10, normalized size = 1. \[ \frac{1}{2} \tan ^{-1}\left (e^{2 x}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[E^(2*x)/(1 + E^(4*x)),x]

[Out]

ArcTan[E^(2*x)]/2

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Maple [A]  time = 0.004, size = 8, normalized size = 0.8 \[{\frac{\arctan \left ( \left ({{\rm e}^{x}} \right ) ^{2} \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(2*x)/(1+exp(4*x)),x)

[Out]

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

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Maxima [A]  time = 0.857471, size = 9, normalized size = 0.9 \[ \frac{1}{2} \, \arctan \left (e^{\left (2 \, x\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(2*x)/(e^(4*x) + 1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.261047, size = 9, normalized size = 0.9 \[ \frac{1}{2} \, \arctan \left (e^{\left (2 \, x\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(2*x)/(e^(4*x) + 1),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.099873, size = 17, normalized size = 1.7 \[ \operatorname{RootSum}{\left (16 z^{2} + 1, \left ( i \mapsto i \log{\left (4 i + e^{2 x} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(2*x)/(1+exp(4*x)),x)

[Out]

RootSum(16*_z**2 + 1, Lambda(_i, _i*log(4*_i + exp(2*x))))

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GIAC/XCAS [A]  time = 0.224156, size = 9, normalized size = 0.9 \[ \frac{1}{2} \, \arctan \left (e^{\left (2 \, x\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(2*x)/(e^(4*x) + 1),x, algorithm="giac")

[Out]

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