3.639 \(\int \frac{e^x}{3-4 e^{2 x}} \, dx\)

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

[Out]

ArcTanh[(2*E^x)/Sqrt[3]]/(2*Sqrt[3])

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Rubi [A]  time = 0.0217914, antiderivative size = 20, normalized size of antiderivative = 1., 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, 206} \[ \frac{\tanh ^{-1}\left (\frac{2 e^x}{\sqrt{3}}\right )}{2 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[E^x/(3 - 4*E^(2*x)),x]

[Out]

ArcTanh[(2*E^x)/Sqrt[3]]/(2*Sqrt[3])

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 206

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

Rubi steps

\begin{align*} \int \frac{e^x}{3-4 e^{2 x}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{3-4 x^2} \, dx,x,e^x\right )\\ &=\frac{\tanh ^{-1}\left (\frac{2 e^x}{\sqrt{3}}\right )}{2 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0063098, size = 20, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\frac{2 e^x}{\sqrt{3}}\right )}{2 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^x/(3 - 4*E^(2*x)),x]

[Out]

ArcTanh[(2*E^x)/Sqrt[3]]/(2*Sqrt[3])

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Maple [A]  time = 0.019, size = 14, normalized size = 0.7 \begin{align*}{\frac{\sqrt{3}}{6}{\it Artanh} \left ({\frac{2\,{{\rm e}^{x}}\sqrt{3}}{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)/(3-4*exp(2*x)),x)

[Out]

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

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Maxima [A]  time = 1.45953, size = 35, normalized size = 1.75 \begin{align*} -\frac{1}{12} \, \sqrt{3} \log \left (-\frac{\sqrt{3} - 2 \, e^{x}}{\sqrt{3} + 2 \, e^{x}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)/(3-4*exp(2*x)),x, algorithm="maxima")

[Out]

-1/12*sqrt(3)*log(-(sqrt(3) - 2*e^x)/(sqrt(3) + 2*e^x))

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Fricas [B]  time = 0.708965, size = 90, normalized size = 4.5 \begin{align*} \frac{1}{12} \, \sqrt{3} \log \left (\frac{4 \, \sqrt{3} e^{x} + 4 \, e^{\left (2 \, x\right )} + 3}{4 \, e^{\left (2 \, x\right )} - 3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)/(3-4*exp(2*x)),x, algorithm="fricas")

[Out]

1/12*sqrt(3)*log((4*sqrt(3)*e^x + 4*e^(2*x) + 3)/(4*e^(2*x) - 3))

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Sympy [A]  time = 0.116251, size = 15, normalized size = 0.75 \begin{align*} \operatorname{RootSum}{\left (48 z^{2} - 1, \left ( i \mapsto i \log{\left (6 i + e^{x} \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)/(3-4*exp(2*x)),x)

[Out]

RootSum(48*_z**2 - 1, Lambda(_i, _i*log(6*_i + exp(x))))

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Giac [B]  time = 1.31441, size = 41, normalized size = 2.05 \begin{align*} \frac{1}{12} \, \sqrt{3} \log \left (\frac{1}{2} \, \sqrt{3} + e^{x}\right ) - \frac{1}{12} \, \sqrt{3} \log \left ({\left | -\frac{1}{2} \, \sqrt{3} + e^{x} \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)/(3-4*exp(2*x)),x, algorithm="giac")

[Out]

1/12*sqrt(3)*log(1/2*sqrt(3) + e^x) - 1/12*sqrt(3)*log(abs(-1/2*sqrt(3) + e^x))