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]

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

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Rubi [A]  time = 0.02, antiderivative size = 20, 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, 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 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])

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}{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*}

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Mathematica [A]  time = 0.01, size = 20, normalized size = 1.00 \[ \frac {\tanh ^{-1}\left (\frac {2 e^x}{\sqrt {3}}\right )}{2 \sqrt {3}} \]

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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fricas [B]  time = 0.41, size = 32, normalized size = 1.60 \[ \frac {1}{12} \, \sqrt {3} \log \left (\frac {4 \, \sqrt {3} e^{x} + 4 \, e^{\left (2 \, x\right )} + 3}{4 \, e^{\left (2 \, x\right )} - 3}\right ) \]

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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giac [B]  time = 0.21, size = 30, normalized size = 1.50 \[ \frac {1}{12} \, \sqrt {3} \log \left (\frac {1}{2} \, \sqrt {3} + e^{x}\right ) - \frac {1}{12} \, \sqrt {3} \log \left ({\left | -\frac {1}{2} \, \sqrt {3} + e^{x} \right |}\right ) \]

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))

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maple [A]  time = 0.03, size = 14, normalized size = 0.70 \[ \frac {\sqrt {3}\, \arctanh \left (\frac {2 \sqrt {3}\, {\mathrm e}^{x}}{3}\right )}{6} \]

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 = 2.32, size = 26, normalized size = 1.30 \[ -\frac {1}{12} \, \sqrt {3} \log \left (-\frac {\sqrt {3} - 2 \, e^{x}}{\sqrt {3} + 2 \, e^{x}}\right ) \]

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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mupad [B]  time = 0.16, size = 13, normalized size = 0.65 \[ \frac {\sqrt {3}\,\mathrm {atanh}\left (\frac {2\,\sqrt {3}\,{\mathrm {e}}^x}{3}\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(x)/(4*exp(2*x) - 3),x)

[Out]

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

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sympy [A]  time = 0.12, size = 15, normalized size = 0.75 \[ \operatorname {RootSum} {\left (48 z^{2} - 1, \left (i \mapsto i \log {\left (6 i + e^{x} \right )} \right )\right )} \]

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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