3.532 \(\int \frac{e^{3 x/4}}{(-2+e^{3 x/4}) \sqrt{-2+e^{3 x/4}+e^{3 x/2}}} \, dx\)

Optimal. Leaf size=40 \[ \frac{2}{3} \tanh ^{-1}\left (\frac{2-5 e^{3 x/4}}{4 \sqrt{e^{3 x/4}+e^{3 x/2}-2}}\right ) \]

[Out]

(2*ArcTanh[(2 - 5*E^((3*x)/4))/(4*Sqrt[-2 + E^((3*x)/4) + E^((3*x)/2)])])/3

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Rubi [A]  time = 0.102506, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2282, 724, 206} \[ \frac{2}{3} \tanh ^{-1}\left (\frac{2-5 e^{3 x/4}}{4 \sqrt{e^{3 x/4}+e^{3 x/2}-2}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[E^((3*x)/4)/((-2 + E^((3*x)/4))*Sqrt[-2 + E^((3*x)/4) + E^((3*x)/2)]),x]

[Out]

(2*ArcTanh[(2 - 5*E^((3*x)/4))/(4*Sqrt[-2 + E^((3*x)/4) + E^((3*x)/2)])])/3

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

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^{3 x/4}}{\left (-2+e^{3 x/4}\right ) \sqrt{-2+e^{3 x/4}+e^{3 x/2}}} \, dx &=\frac{4}{3} \operatorname{Subst}\left (\int \frac{1}{(-2+x) \sqrt{-2+x+x^2}} \, dx,x,e^{3 x/4}\right )\\ &=-\left (\frac{8}{3} \operatorname{Subst}\left (\int \frac{1}{16-x^2} \, dx,x,\frac{-2+5 e^{3 x/4}}{\sqrt{-2+e^{3 x/4}+e^{3 x/2}}}\right )\right )\\ &=\frac{2}{3} \tanh ^{-1}\left (\frac{2-5 e^{3 x/4}}{4 \sqrt{-2+e^{3 x/4}+e^{3 x/2}}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0196298, size = 40, normalized size = 1. \[ -\frac{2}{3} \tanh ^{-1}\left (\frac{5 e^{3 x/4}-2}{4 \sqrt{e^{3 x/4}+e^{3 x/2}-2}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^((3*x)/4)/((-2 + E^((3*x)/4))*Sqrt[-2 + E^((3*x)/4) + E^((3*x)/2)]),x]

[Out]

(-2*ArcTanh[(-2 + 5*E^((3*x)/4))/(4*Sqrt[-2 + E^((3*x)/4) + E^((3*x)/2)])])/3

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Maple [F]  time = 0.019, size = 0, normalized size = 0. \begin{align*} \int{{{\rm e}^{{\frac{3\,x}{4}}}} \left ( -2+{{\rm e}^{{\frac{3\,x}{4}}}} \right ) ^{-1}{\frac{1}{\sqrt{-2+{{\rm e}^{{\frac{3\,x}{4}}}}+{{\rm e}^{{\frac{3\,x}{2}}}}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(3/4*x)/(-2+exp(3/4*x))/(-2+exp(3/4*x)+exp(3/2*x))^(1/2),x)

[Out]

int(exp(3/4*x)/(-2+exp(3/4*x))/(-2+exp(3/4*x)+exp(3/2*x))^(1/2),x)

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Maxima [A]  time = 1.41629, size = 53, normalized size = 1.32 \begin{align*} -\frac{2}{3} \, \log \left (\frac{4 \, \sqrt{e^{\left (\frac{3}{2} \, x\right )} + e^{\left (\frac{3}{4} \, x\right )} - 2}}{{\left | e^{\left (\frac{3}{4} \, x\right )} - 2 \right |}} + \frac{8}{{\left | e^{\left (\frac{3}{4} \, x\right )} - 2 \right |}} + 5\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(3/4*x)/(-2+exp(3/4*x))/(-2+exp(3/4*x)+exp(3/2*x))^(1/2),x, algorithm="maxima")

[Out]

-2/3*log(4*sqrt(e^(3/2*x) + e^(3/4*x) - 2)/abs(e^(3/4*x) - 2) + 8/abs(e^(3/4*x) - 2) + 5)

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Fricas [A]  time = 1.9534, size = 154, normalized size = 3.85 \begin{align*} -\frac{2}{3} \, \log \left (\sqrt{e^{\left (\frac{3}{2} \, x\right )} + e^{\left (\frac{3}{4} \, x\right )} - 2} - e^{\left (\frac{3}{4} \, x\right )} + 4\right ) + \frac{2}{3} \, \log \left (\sqrt{e^{\left (\frac{3}{2} \, x\right )} + e^{\left (\frac{3}{4} \, x\right )} - 2} - e^{\left (\frac{3}{4} \, x\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(3/4*x)/(-2+exp(3/4*x))/(-2+exp(3/4*x)+exp(3/2*x))^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\frac{3 x}{4}}}{\left (e^{\frac{3 x}{4}} - 2\right ) \sqrt{e^{\frac{3 x}{4}} + e^{\frac{3 x}{2}} - 2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(3/4*x)/(-2+exp(3/4*x))/(-2+exp(3/4*x)+exp(3/2*x))**(1/2),x)

[Out]

Integral(exp(3*x/4)/((exp(3*x/4) - 2)*sqrt(exp(3*x/4) + exp(3*x/2) - 2)), x)

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Giac [A]  time = 1.20086, size = 65, normalized size = 1.62 \begin{align*} -\frac{2}{3} \, \log \left ({\left | \sqrt{e^{\left (\frac{3}{2} \, x\right )} + e^{\left (\frac{3}{4} \, x\right )} - 2} - e^{\left (\frac{3}{4} \, x\right )} + 4 \right |}\right ) + \frac{2}{3} \, \log \left ({\left | \sqrt{e^{\left (\frac{3}{2} \, x\right )} + e^{\left (\frac{3}{4} \, x\right )} - 2} - e^{\left (\frac{3}{4} \, x\right )} \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(3/4*x)/(-2+exp(3/4*x))/(-2+exp(3/4*x)+exp(3/2*x))^(1/2),x, algorithm="giac")

[Out]

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