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

Optimal. Leaf size=18 \[ \frac {\tan ^{-1}\left (\sqrt {\frac {3}{2}} e^x\right )}{\sqrt {6}} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2249, 203} \[ \frac {\tan ^{-1}\left (\sqrt {\frac {3}{2}} e^x\right )}{\sqrt {6}} \]

Antiderivative was successfully verified.

[In]

Int[E^x/(2 + 3*E^(2*x)),x]

[Out]

ArcTan[Sqrt[3/2]*E^x]/Sqrt[6]

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])

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]

Rubi steps

\begin {align*} \int \frac {e^x}{2+3 e^{2 x}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{2+3 x^2} \, dx,x,e^x\right )\\ &=\frac {\tan ^{-1}\left (\sqrt {\frac {3}{2}} e^x\right )}{\sqrt {6}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 18, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\sqrt {\frac {3}{2}} e^x\right )}{\sqrt {6}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^x/(2 + 3*E^(2*x)),x]

[Out]

ArcTan[Sqrt[3/2]*E^x]/Sqrt[6]

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fricas [A]  time = 0.42, size = 13, normalized size = 0.72 \[ \frac {1}{6} \, \sqrt {6} \arctan \left (\frac {1}{2} \, \sqrt {6} e^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.11, size = 13, normalized size = 0.72 \[ \frac {1}{6} \, \sqrt {6} \arctan \left (\frac {1}{2} \, \sqrt {6} e^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 14, normalized size = 0.78 \[ \frac {\sqrt {6}\, \arctan \left (\frac {\sqrt {6}\, {\mathrm e}^{x}}{2}\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.33, size = 13, normalized size = 0.72 \[ \frac {1}{6} \, \sqrt {6} \arctan \left (\frac {1}{2} \, \sqrt {6} e^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.11, size = 13, normalized size = 0.72 \[ \frac {\sqrt {6}\,\mathrm {atan}\left (\frac {\sqrt {6}\,{\mathrm {e}}^x}{2}\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)/(3*exp(2*x) + 2),x)

[Out]

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

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sympy [A]  time = 0.11, size = 15, normalized size = 0.83 \[ \operatorname {RootSum} {\left (24 z^{2} + 1, \left (i \mapsto i \log {\left (4 i + e^{x} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(24*_z**2 + 1, Lambda(_i, _i*log(4*_i + exp(x))))

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