3.2890 \(\int \frac{1}{\sqrt{-1+x} \sqrt{-12+8 x-x^2}} \, dx\)

Optimal. Leaf size=25 \[ -\frac{2 \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{6-x}}{2}\right ),\frac{4}{5}\right )}{\sqrt{5}} \]

[Out]

(-2*EllipticF[ArcSin[Sqrt[6 - x]/2], 4/5])/Sqrt[5]

________________________________________________________________________________________

Rubi [A]  time = 0.0163658, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {718, 419} \[ -\frac{2 F\left (\sin ^{-1}\left (\frac{\sqrt{6-x}}{2}\right )|\frac{4}{5}\right )}{\sqrt{5}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[-1 + x]*Sqrt[-12 + 8*x - x^2]),x]

[Out]

(-2*EllipticF[ArcSin[Sqrt[6 - x]/2], 4/5])/Sqrt[5]

Rule 718

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

Rule 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{-1+x} \sqrt{-12+8 x-x^2}} \, dx &=-\frac{2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1-\frac{4 x^2}{5}}} \, dx,x,\frac{\sqrt{12-2 x}}{2 \sqrt{2}}\right )}{\sqrt{5}}\\ &=-\frac{2 F\left (\sin ^{-1}\left (\frac{\sqrt{6-x}}{2}\right )|\frac{4}{5}\right )}{\sqrt{5}}\\ \end{align*}

Mathematica [B]  time = 0.0700341, size = 68, normalized size = 2.72 \[ -\frac{2 \sqrt{\frac{x-6}{x-1}} \sqrt{\frac{x-2}{x-1}} (x-1) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{5}}{\sqrt{x-1}}\right ),\frac{1}{5}\right )}{\sqrt{5} \sqrt{-x^2+8 x-12}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[-1 + x]*Sqrt[-12 + 8*x - x^2]),x]

[Out]

(-2*Sqrt[(-6 + x)/(-1 + x)]*Sqrt[(-2 + x)/(-1 + x)]*(-1 + x)*EllipticF[ArcSin[Sqrt[5]/Sqrt[-1 + x]], 1/5])/(Sq
rt[5]*Sqrt[-12 + 8*x - x^2])

________________________________________________________________________________________

Maple [B]  time = 0.013, size = 55, normalized size = 2.2 \begin{align*}{\frac{2\,\sqrt{5}}{5\,{x}^{2}-40\,x+60}{\it EllipticF} \left ({\frac{1}{2}\sqrt{6-x}},{\frac{2\,\sqrt{5}}{5}} \right ) \sqrt{-2+x}\sqrt{6-x}\sqrt{-{x}^{2}+8\,x-12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-1+x)^(1/2)/(-x^2+8*x-12)^(1/2),x)

[Out]

2/5*EllipticF(1/2*(6-x)^(1/2),2/5*5^(1/2))*(-2+x)^(1/2)*5^(1/2)*(6-x)^(1/2)*(-x^2+8*x-12)^(1/2)/(x^2-8*x+12)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-x^{2} + 8 \, x - 12} \sqrt{x - 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-1+x)^(1/2)/(-x^2+8*x-12)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-x^2 + 8*x - 12)*sqrt(x - 1)), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-x^{2} + 8 \, x - 12} \sqrt{x - 1}}{x^{3} - 9 \, x^{2} + 20 \, x - 12}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-1+x)^(1/2)/(-x^2+8*x-12)^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-x^2 + 8*x - 12)*sqrt(x - 1)/(x^3 - 9*x^2 + 20*x - 12), x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- \left (x - 6\right ) \left (x - 2\right )} \sqrt{x - 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-1+x)**(1/2)/(-x**2+8*x-12)**(1/2),x)

[Out]

Integral(1/(sqrt(-(x - 6)*(x - 2))*sqrt(x - 1)), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-x^{2} + 8 \, x - 12} \sqrt{x - 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-1+x)^(1/2)/(-x^2+8*x-12)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-x^2 + 8*x - 12)*sqrt(x - 1)), x)