3.29.90 \(\int \frac {1}{\sqrt {-1+x} \sqrt {-12+8 x-x^2}} \, dx\) [2890]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, 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 = {732, 430} \begin {gather*} -\frac {2 F\left (\text {ArcSin}\left (\frac {\sqrt {6-x}}{2}\right )|\frac {4}{5}\right )}{\sqrt {5}} \end {gather*}

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 430

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

Rule 732

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]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {-1+x} \sqrt {-12+8 x-x^2}} \, dx &=-\frac {2 \text {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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(68\) vs. \(2(25)=50\).
time = 30.07, size = 68, normalized size = 2.72 \begin {gather*} -\frac {2 \sqrt {\frac {-6+x}{-1+x}} \sqrt {\frac {-2+x}{-1+x}} (-1+x) F\left (\sin ^{-1}\left (\frac {\sqrt {5}}{\sqrt {-1+x}}\right )|\frac {1}{5}\right )}{\sqrt {5} \sqrt {-12+8 x-x^2}} \end {gather*}

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])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(49\) vs. \(2(20)=40\).
time = 0.13, size = 50, normalized size = 2.00

method	result	size
default	\(-\frac {2 \EllipticF \left (\sqrt {-1+x}, \frac {\sqrt {5}}{5}\right ) \sqrt {2-x}\, \sqrt {30-5 x}\, \sqrt {-x^{2}+8 x -12}}{5 \left (x^{2}-8 x +12\right )}\)	\(50\)
elliptic	\(\frac {2 \sqrt {-\left (-1+x \right ) \left (x^{2}-8 x +12\right )}\, \sqrt {30-5 x}\, \sqrt {2-x}\, \EllipticF \left (\sqrt {-1+x}, \frac {\sqrt {5}}{5}\right )}{5 \sqrt {-x^{2}+8 x -12}\, \sqrt {-x^{3}+9 x^{2}-20 x +12}}\)	\(72\)

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,method=_RETURNVERBOSE)

[Out]

-2/5*EllipticF((-1+x)^(1/2),1/5*5^(1/2))*(2-x)^(1/2)*(30-5*x)^(1/2)*(-x^2+8*x-12)^(1/2)/(x^2-8*x+12)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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]

0

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {- \left (x - 6\right ) \left (x - 2\right )} \sqrt {x - 1}}\, dx \end {gather*}

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)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {1}{\sqrt {x-1}\,\sqrt {-x^2+8\,x-12}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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