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

Optimal. Leaf size=20 \[ \sqrt{\frac{2}{3}} \tan ^{-1}\left (\sqrt{\frac{3}{2}} e^x\right ) \]

[Out]

Sqrt[2/3]*ArcTan[Sqrt[3/2]*E^x]

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Rubi [A]  time = 0.0226488, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {12, 2249, 203} \[ \sqrt{\frac{2}{3}} \tan ^{-1}\left (\sqrt{\frac{3}{2}} e^x\right ) \]

Antiderivative was successfully verified.

[In]

Int[(2*E^x)/(2 + 3*E^(2*x)),x]

[Out]

Sqrt[2/3]*ArcTan[Sqrt[3/2]*E^x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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{2 e^x}{2+3 e^{2 x}} \, dx &=2 \int \frac{e^x}{2+3 e^{2 x}} \, dx\\ &=2 \operatorname{Subst}\left (\int \frac{1}{2+3 x^2} \, dx,x,e^x\right )\\ &=\sqrt{\frac{2}{3}} \tan ^{-1}\left (\sqrt{\frac{3}{2}} e^x\right )\\ \end{align*}

Mathematica [A]  time = 0.0051819, size = 20, normalized size = 1. \[ \sqrt{\frac{2}{3}} \tan ^{-1}\left (\sqrt{\frac{3}{2}} e^x\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(2*E^x)/(2 + 3*E^(2*x)),x]

[Out]

Sqrt[2/3]*ArcTan[Sqrt[3/2]*E^x]

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Maple [A]  time = 0., size = 14, normalized size = 0.7 \begin{align*}{\frac{\sqrt{6}}{3}\arctan \left ({\frac{{{\rm e}^{x}}\sqrt{6}}{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*exp(x)/(2+3*exp(2*x)),x)

[Out]

1/3*arctan(1/2*exp(x)*6^(1/2))*6^(1/2)

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Maxima [A]  time = 1.421, size = 18, normalized size = 0.9 \begin{align*} \frac{1}{3} \, \sqrt{6} \arctan \left (\frac{1}{2} \, \sqrt{6} e^{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*exp(x)/(2+3*exp(2*x)),x, algorithm="maxima")

[Out]

1/3*sqrt(6)*arctan(1/2*sqrt(6)*e^x)

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Fricas [A]  time = 1.92223, size = 72, normalized size = 3.6 \begin{align*} \frac{1}{3} \, \sqrt{3} \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{3} \sqrt{2} e^{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*exp(x)/(2+3*exp(2*x)),x, algorithm="fricas")

[Out]

1/3*sqrt(3)*sqrt(2)*arctan(1/2*sqrt(3)*sqrt(2)*e^x)

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Sympy [A]  time = 0.106325, size = 15, normalized size = 0.75 \begin{align*} \operatorname{RootSum}{\left (6 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(2*exp(x)/(2+3*exp(2*x)),x)

[Out]

RootSum(6*_z**2 + 1, Lambda(_i, _i*log(2*_i + exp(x))))

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Giac [A]  time = 1.06551, size = 18, normalized size = 0.9 \begin{align*} \frac{1}{3} \, \sqrt{6} \arctan \left (\frac{1}{2} \, \sqrt{6} e^{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*exp(x)/(2+3*exp(2*x)),x, algorithm="giac")

[Out]

1/3*sqrt(6)*arctan(1/2*sqrt(6)*e^x)