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

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 = 0.0193594, 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, Rules used = {2249, 203} \[ \frac{1}{2} \tan ^{-1}\left (\frac{e^x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[E^x/(4 + E^(2*x)),x]

[Out]

ArcTan[E^x/2]/2

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}{4+e^{2 x}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{4+x^2} \, dx,x,e^x\right )\\ &=\frac{1}{2} \tan ^{-1}\left (\frac{e^x}{2}\right )\\ \end{align*}

Mathematica [A]  time = 0.0026816, size = 12, normalized size = 1. \[ \frac{1}{2} \tan ^{-1}\left (\frac{e^x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^x/(4 + E^(2*x)),x]

[Out]

ArcTan[E^x/2]/2

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Maple [A]  time = 0.02, size = 8, normalized size = 0.7 \begin{align*}{\frac{1}{2}\arctan \left ({\frac{{{\rm e}^{x}}}{2}} \right ) } \end{align*}

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 = 1.45997, size = 9, normalized size = 0.75 \begin{align*} \frac{1}{2} \, \arctan \left (\frac{1}{2} \, e^{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)/(4+exp(2*x)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.810531, size = 28, normalized size = 2.33 \begin{align*} \frac{1}{2} \, \arctan \left (\frac{1}{2} \, e^{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)/(4+exp(2*x)),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.107225, size = 15, normalized size = 1.25 \begin{align*} \operatorname{RootSum}{\left (16 z^{2} + 1, \left ( i \mapsto i \log{\left (8 i + e^{x} \right )} \right )\right )} \end{align*}

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 [A]  time = 1.23667, size = 9, normalized size = 0.75 \begin{align*} \frac{1}{2} \, \arctan \left (\frac{1}{2} \, e^{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)/(4+exp(2*x)),x, algorithm="giac")

[Out]

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