3.766 \(\int \frac {1}{\sqrt {8 x-8 x^2+4 x^3-x^4}} \, dx\)

Optimal. Leaf size=17 \[ -\frac {F\left (\sin ^{-1}(1-x)|-\frac {1}{3}\right )}{\sqrt {3}} \]

[Out]

1/3*EllipticF(-1+x,1/3*I*3^(1/2))*3^(1/2)

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Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {1106, 1095, 419} \[ -\frac {F\left (\sin ^{-1}(1-x)|-\frac {1}{3}\right )}{\sqrt {3}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[8*x - 8*x^2 + 4*x^3 - x^4],x]

[Out]

-(EllipticF[ArcSin[1 - x], -1/3]/Sqrt[3])

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)])

Rule 1095

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[2*Sqrt[-c], I
nt[1/(Sqrt[b + q + 2*c*x^2]*Sqrt[-b + q - 2*c*x^2]), x], x]] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0] &&
LtQ[c, 0]

Rule 1106

Int[(P4_)^(p_), x_Symbol] :> With[{a = Coeff[P4, x, 0], b = Coeff[P4, x, 1], c = Coeff[P4, x, 2], d = Coeff[P4
, x, 3], e = Coeff[P4, x, 4]}, Subst[Int[SimplifyIntegrand[(a + d^4/(256*e^3) - (b*d)/(8*e) + (c - (3*d^2)/(8*
e))*x^2 + e*x^4)^p, x], x], x, d/(4*e) + x] /; EqQ[d^3 - 4*c*d*e + 8*b*e^2, 0] && NeQ[d, 0]] /; FreeQ[p, x] &&
PolyQ[P4, x, 4] && NeQ[p, 2] && NeQ[p, 3]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {8 x-8 x^2+4 x^3-x^4}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\sqrt {3-2 x^2-x^4}} \, dx,x,-1+x\right )\\ &=2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {2-2 x^2} \sqrt {6+2 x^2}} \, dx,x,-1+x\right )\\ &=-\frac {F\left (\sin ^{-1}(1-x)|-\frac {1}{3}\right )}{\sqrt {3}}\\ \end {align*}

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Mathematica [C]  time = 0.15, size = 156, normalized size = 9.18 \[ \frac {\sqrt {\frac {4 i}{x}+\sqrt {3}-i} \sqrt {-\frac {i (x-2)}{\left (\sqrt {3}-i\right ) x}} x \left (-i \sqrt {3} x+x-4\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {\sqrt {3}+i-\frac {4 i}{x}}}{\sqrt {2} \sqrt [4]{3}}\right )|\frac {2 \sqrt {3}}{-i+\sqrt {3}}\right )}{\sqrt {2} \sqrt {-\frac {4 i}{x}+\sqrt {3}+i} \sqrt {-x \left (x^3-4 x^2+8 x-8\right )}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/Sqrt[8*x - 8*x^2 + 4*x^3 - x^4],x]

[Out]

(Sqrt[-I + Sqrt[3] + (4*I)/x]*Sqrt[((-I)*(-2 + x))/((-I + Sqrt[3])*x)]*x*(-4 + x - I*Sqrt[3]*x)*EllipticF[ArcS
in[Sqrt[I + Sqrt[3] - (4*I)/x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-I + Sqrt[3])])/(Sqrt[2]*Sqrt[I + Sqrt[3] - (4
*I)/x]*Sqrt[-(x*(-8 + 8*x - 4*x^2 + x^3))])

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fricas [F]  time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x}}{x^{4} - 4 \, x^{3} + 8 \, x^{2} - 8 \, x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^4+4*x^3-8*x^2+8*x)^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-x^4 + 4*x^3 - 8*x^2 + 8*x)/(x^4 - 4*x^3 + 8*x^2 - 8*x), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^4+4*x^3-8*x^2+8*x)^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(-x^4 + 4*x^3 - 8*x^2 + 8*x), x)

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maple [B]  time = 0.07, size = 200, normalized size = 11.76 \[ \frac {2 \left (-1-i \sqrt {3}\right ) \sqrt {\frac {\left (i \sqrt {3}-1\right ) x}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}\, \left (x -2\right )^{2} \sqrt {\frac {x -1+i \sqrt {3}}{\left (1-i \sqrt {3}\right ) \left (x -2\right )}}\, \sqrt {\frac {x -1-i \sqrt {3}}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}\, \EllipticF \left (\sqrt {\frac {\left (i \sqrt {3}-1\right ) x}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}, \sqrt {\frac {\left (1+i \sqrt {3}\right ) \left (-1-i \sqrt {3}\right )}{\left (i \sqrt {3}-1\right ) \left (1-i \sqrt {3}\right )}}\right )}{\left (i \sqrt {3}-1\right ) \sqrt {-\left (x -2\right ) \left (x -1+i \sqrt {3}\right ) \left (x -1-i \sqrt {3}\right ) x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-x^4+4*x^3-8*x^2+8*x)^(1/2),x)

[Out]

2*(-1-I*3^(1/2))*((I*3^(1/2)-1)/(1+I*3^(1/2))/(x-2)*x)^(1/2)*(x-2)^2*((x-1+I*3^(1/2))/(1-I*3^(1/2))/(x-2))^(1/
2)*((x-1-I*3^(1/2))/(1+I*3^(1/2))/(x-2))^(1/2)/(I*3^(1/2)-1)/(-(x-2)*(x-1+I*3^(1/2))*(x-1-I*3^(1/2))*x)^(1/2)*
EllipticF(((I*3^(1/2)-1)/(1+I*3^(1/2))/(x-2)*x)^(1/2),((1+I*3^(1/2))*(-1-I*3^(1/2))/(I*3^(1/2)-1)/(1-I*3^(1/2)
))^(1/2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^4+4*x^3-8*x^2+8*x)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(-x^4 + 4*x^3 - 8*x^2 + 8*x), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.06 \[ \int \frac {1}{\sqrt {-x^4+4\,x^3-8\,x^2+8\,x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(8*x - 8*x^2 + 4*x^3 - x^4)^(1/2),x)

[Out]

int(1/(8*x - 8*x^2 + 4*x^3 - x^4)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {- x^{4} + 4 x^{3} - 8 x^{2} + 8 x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x**4+4*x**3-8*x**2+8*x)**(1/2),x)

[Out]

Integral(1/sqrt(-x**4 + 4*x**3 - 8*x**2 + 8*x), x)

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