3.101 \(\int \frac{e^{6 x}}{1+e^{4 x}} \, dx\)

Optimal. Leaf size=20 \[ \frac{e^{2 x}}{2}-\frac{1}{2} \tan ^{-1}\left (e^{2 x}\right ) \]

[Out]

E^(2*x)/2 - ArcTan[E^(2*x)]/2

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Rubi [A]  time = 0.0396088, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{e^{2 x}}{2}-\frac{1}{2} \tan ^{-1}\left (e^{2 x}\right ) \]

Antiderivative was successfully verified.

[In]  Int[E^(6*x)/(1 + E^(4*x)),x]

[Out]

E^(2*x)/2 - ArcTan[E^(2*x)]/2

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Rubi in Sympy [A]  time = 4.07059, size = 14, normalized size = 0.7 \[ \frac{e^{2 x}}{2} - \frac{\operatorname{atan}{\left (e^{2 x} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(6*x)/(1+exp(4*x)),x)

[Out]

exp(2*x)/2 - atan(exp(2*x))/2

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Mathematica [A]  time = 0.00885969, size = 20, normalized size = 1. \[ \frac{e^{2 x}}{2}-\frac{1}{2} \tan ^{-1}\left (e^{2 x}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[E^(6*x)/(1 + E^(4*x)),x]

[Out]

E^(2*x)/2 - ArcTan[E^(2*x)]/2

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Maple [A]  time = 0.006, size = 15, normalized size = 0.8 \[{\frac{ \left ({{\rm e}^{x}} \right ) ^{2}}{2}}-{\frac{\arctan \left ( \left ({{\rm e}^{x}} \right ) ^{2} \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(6*x)/(1+exp(4*x)),x)

[Out]

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

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Maxima [A]  time = 1.48415, size = 19, normalized size = 0.95 \[ -\frac{1}{2} \, \arctan \left (e^{\left (2 \, x\right )}\right ) + \frac{1}{2} \, e^{\left (2 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(6*x)/(e^(4*x) + 1),x, algorithm="maxima")

[Out]

-1/2*arctan(e^(2*x)) + 1/2*e^(2*x)

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Fricas [A]  time = 0.222383, size = 19, normalized size = 0.95 \[ -\frac{1}{2} \, \arctan \left (e^{\left (2 \, x\right )}\right ) + \frac{1}{2} \, e^{\left (2 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(6*x)/(e^(4*x) + 1),x, algorithm="fricas")

[Out]

-1/2*arctan(e^(2*x)) + 1/2*e^(2*x)

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Sympy [A]  time = 0.120295, size = 24, normalized size = 1.2 \[ \frac{e^{2 x}}{2} + \operatorname{RootSum}{\left (16 z^{2} + 1, \left ( i \mapsto i \log{\left (- 4 i + e^{2 x} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(6*x)/(1+exp(4*x)),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.201267, size = 19, normalized size = 0.95 \[ -\frac{1}{2} \, \arctan \left (e^{\left (2 \, x\right )}\right ) + \frac{1}{2} \, e^{\left (2 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(6*x)/(e^(4*x) + 1),x, algorithm="giac")

[Out]

-1/2*arctan(e^(2*x)) + 1/2*e^(2*x)