3.1.99 \(\int \frac {2 e^x}{2+3 e^{2 x}} \, dx\) [99]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.00, 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, 2281, 209} \begin {gather*} \sqrt {\frac {2}{3}} \text {ArcTan}\left (\sqrt {\frac {3}{2}} e^x\right ) \end {gather*}

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 209

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

Rule 2281

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]

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 \text {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*}

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Mathematica [A]
time = 0.00, size = 20, normalized size = 1.00 \begin {gather*} \sqrt {\frac {2}{3}} \tan ^{-1}\left (\sqrt {\frac {3}{2}} e^x\right ) \end {gather*}

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 = 0.01, size = 14, normalized size = 0.70

method	result	size
default	\(\frac {\arctan \left (\frac {{\mathrm e}^{x} \sqrt {6}}{2}\right ) \sqrt {6}}{3}\)	\(14\)
risch	\(\frac {i \sqrt {6}\, \ln \left ({\mathrm e}^{x}+\frac {i \sqrt {6}}{3}\right )}{6}-\frac {i \sqrt {6}\, \ln \left ({\mathrm e}^{x}-\frac {i \sqrt {6}}{3}\right )}{6}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*exp(x)/(2+3*exp(2*x)),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 2.02, size = 13, normalized size = 0.65 \begin {gather*} \frac {1}{3} \, \sqrt {6} \arctan \left (\frac {1}{2} \, \sqrt {6} e^{x}\right ) \end {gather*}

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 = 0.39, size = 19, normalized size = 0.95 \begin {gather*} \frac {1}{3} \, \sqrt {3} \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {3} \sqrt {2} e^{x}\right ) \end {gather*}

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 = 0.04, size = 15, normalized size = 0.75 \begin {gather*} \operatorname {RootSum} {\left (6 z^{2} + 1, \left ( i \mapsto i \log {\left (2 i + e^{x} \right )} \right )\right )} \end {gather*}

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 = 1.24, size = 13, normalized size = 0.65 \begin {gather*} \frac {1}{3} \, \sqrt {6} \arctan \left (\frac {1}{2} \, \sqrt {6} e^{x}\right ) \end {gather*}

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)

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Mupad [B]
time = 0.09, size = 13, normalized size = 0.65 \begin {gather*} \frac {\sqrt {6}\,\mathrm {atan}\left (\frac {\sqrt {6}\,{\mathrm {e}}^x}{2}\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*exp(x))/(3*exp(2*x) + 2),x)

[Out]

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

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