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

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

[Out]

E^x - ArcTan[E^x]

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Rubi [A]  time = 0.039652, antiderivative size = 10, 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 \[ e^x-\tan ^{-1}\left (e^x\right ) \]

Antiderivative was successfully verified.

[In]  Int[E^(3*x)/(1 + E^(2*x)),x]

[Out]

E^x - ArcTan[E^x]

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Rubi in Sympy [A]  time = 7.6365, size = 7, normalized size = 0.7 \[ e^{x} - \operatorname{atan}{\left (e^{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

exp(x) - atan(exp(x))

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Mathematica [A]  time = 0.00637822, size = 10, normalized size = 1. \[ e^x-\tan ^{-1}\left (e^x\right ) \]

Antiderivative was successfully verified.

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

[Out]

E^x - ArcTan[E^x]

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Maple [A]  time = 0.005, size = 9, normalized size = 0.9 \[{{\rm e}^{x}}-\arctan \left ({{\rm e}^{x}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(3*x)/(1+exp(2*x)),x)

[Out]

exp(x)-arctan(exp(x))

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Maxima [A]  time = 0.856133, size = 11, normalized size = 1.1 \[ -\arctan \left (e^{x}\right ) + e^{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-arctan(e^x) + e^x

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Fricas [A]  time = 0.298067, size = 11, normalized size = 1.1 \[ -\arctan \left (e^{x}\right ) + e^{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-arctan(e^x) + e^x

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Sympy [A]  time = 0.109671, size = 19, normalized size = 1.9 \[ e^{x} + \operatorname{RootSum}{\left (4 z^{2} + 1, \left ( i \mapsto i \log{\left (- 2 i + e^{x} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.242792, size = 11, normalized size = 1.1 \[ -\arctan \left (e^{x}\right ) + e^{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-arctan(e^x) + e^x