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 \[ \frac {2}{3} \tanh ^{-1}\left (\frac {2-5 e^{3 x/4}}{4 \sqrt {e^{3 x/4}+e^{3 x/2}-2}}\right ) \]

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Rubi [A]  time = 0.10, antiderivative size = 40, normalized size of antiderivative = 1.00, 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 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 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 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]]]

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*}

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Mathematica [A]  time = 0.02, size = 40, normalized size = 1.00 \[ -\frac {2}{3} \tanh ^{-1}\left (\frac {5 e^{3 x/4}-2}{4 \sqrt {e^{3 x/4}+e^{3 x/2}-2}}\right ) \]

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{3 x/4}}{\left (-2+e^{3 x/4}\right ) \sqrt {-2+e^{3 x/4}+e^{3 x/2}}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Could not integrate

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fricas [A]  time = 1.03, size = 46, normalized size = 1.15 \[ -\frac {2}{3} \, \log \left (\sqrt {e^{\left (\frac {3}{2} \, x\right )} + e^{\left (\frac {3}{4} \, x\right )} - 2} - e^{\left (\frac {3}{4} \, x\right )} + 4\right ) + \frac {2}{3} \, \log \left (\sqrt {e^{\left (\frac {3}{2} \, x\right )} + e^{\left (\frac {3}{4} \, x\right )} - 2} - e^{\left (\frac {3}{4} \, x\right )}\right ) \]

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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giac [A]  time = 0.82, size = 48, normalized size = 1.20 \[ -\frac {2}{3} \, \log \left ({\left | \sqrt {e^{\left (\frac {3}{2} \, x\right )} + e^{\left (\frac {3}{4} \, x\right )} - 2} - e^{\left (\frac {3}{4} \, x\right )} + 4 \right |}\right ) + \frac {2}{3} \, \log \left ({\left | \sqrt {e^{\left (\frac {3}{2} \, x\right )} + e^{\left (\frac {3}{4} \, x\right )} - 2} - e^{\left (\frac {3}{4} \, x\right )} \right |}\right ) \]

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)))

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maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{\frac {3 x}{4}}}{\left (-2+{\mathrm e}^{\frac {3 x}{4}}\right ) \sqrt {-2+{\mathrm e}^{\frac {3 x}{4}}+{\mathrm e}^{\frac {3 x}{2}}}}\, dx\]

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 = 1.30, size = 39, normalized size = 0.98 \[ -\frac {2}{3} \, \log \left (\frac {4 \, \sqrt {e^{\left (\frac {3}{2} \, x\right )} + e^{\left (\frac {3}{4} \, x\right )} - 2}}{{\left | e^{\left (\frac {3}{4} \, x\right )} - 2 \right |}} + \frac {8}{{\left | e^{\left (\frac {3}{4} \, x\right )} - 2 \right |}} + 5\right ) \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\mathrm {e}}^{\frac {3\,x}{4}}}{\left ({\mathrm {e}}^{\frac {3\,x}{4}}-2\right )\,\sqrt {{\mathrm {e}}^{\frac {3\,x}{2}}+{\mathrm {e}}^{\frac {3\,x}{4}}-2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{\frac {3 x}{4}}}{\left (e^{\frac {3 x}{4}} - 2\right ) \sqrt {e^{\frac {3 x}{4}} + e^{\frac {3 x}{2}} - 2}}\, dx \]

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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