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

Optimal. Leaf size=4 \[ \tan ^{-1}\left (e^x\right ) \]

[Out]

ArcTan[E^x]

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Rubi [A]  time = 0.0190307, antiderivative size = 4, 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, Rules used = {2249, 203} \[ \tan ^{-1}\left (e^x\right ) \]

Antiderivative was successfully verified.

[In]

Int[E^x/(1 + E^(2*x)),x]

[Out]

ArcTan[E^x]

Rule 2249

Int[((a_) + (b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))))^(p_.)*(G_)^((h_.)*((f_.) + (g_.)*(x_))), x_Symbol] :> Wit
h[{m = FullSimplify[(d*e*Log[F])/(g*h*Log[G])]}, Dist[Denominator[m]/(g*h*Log[G]), Subst[Int[x^(Denominator[m]
- 1)*(a + b*F^(c*e - (d*e*f)/g)*x^Numerator[m])^p, x], x, G^((h*(f + g*x))/Denominator[m])], x] /; LtQ[m, -1]
|| GtQ[m, 1]] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{e^x}{1+e^{2 x}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,e^x\right )\\ &=\tan ^{-1}\left (e^x\right )\\ \end{align*}

Mathematica [A]  time = 0.0025029, size = 4, normalized size = 1. \[ \tan ^{-1}\left (e^x\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^x/(1 + E^(2*x)),x]

[Out]

ArcTan[E^x]

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Maple [A]  time = 0.02, size = 4, normalized size = 1. \begin{align*} \arctan \left ({{\rm e}^{x}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)/(1+exp(2*x)),x)

[Out]

arctan(exp(x))

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Maxima [A]  time = 1.88194, size = 4, normalized size = 1. \begin{align*} \arctan \left (e^{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)/(1+exp(2*x)),x, algorithm="maxima")

[Out]

arctan(e^x)

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Fricas [A]  time = 0.869882, size = 18, normalized size = 4.5 \begin{align*} \arctan \left (e^{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)/(1+exp(2*x)),x, algorithm="fricas")

[Out]

arctan(e^x)

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Sympy [B]  time = 0.104633, size = 15, normalized size = 3.75 \begin{align*} \operatorname{RootSum}{\left (4 z^{2} + 1, \left ( i \mapsto i \log{\left (2 i + e^{x} \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)/(1+exp(2*x)),x)

[Out]

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

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Giac [A]  time = 1.25289, size = 4, normalized size = 1. \begin{align*} \arctan \left (e^{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)/(1+exp(2*x)),x, algorithm="giac")

[Out]

arctan(e^x)