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

Optimal. Leaf size=144 \[ \frac {\sqrt {\sqrt {a+4}+1} \left (\frac {(x-1)^2}{1-\sqrt {a+4}}+1\right ) F\left (\tan ^{-1}\left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right )|-\frac {2 \sqrt {a+4}}{1-\sqrt {a+4}}\right )}{\sqrt {\frac {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}{\frac {(x-1)^2}{\sqrt {a+4}+1}+1}} \sqrt {a-(x-1)^4-2 (x-1)^2+3}} \]

[Out]

(1/(1+(-1+x)^2/(1+(4+a)^(1/2))))^(1/2)*(1+(-1+x)^2/(1+(4+a)^(1/2)))^(1/2)*EllipticF((-1+x)/(1+(4+a)^(1/2))^(1/
2)/(1+(-1+x)^2/(1+(4+a)^(1/2)))^(1/2),(-2*(4+a)^(1/2)/(1-(4+a)^(1/2)))^(1/2))*(1+(-1+x)^2/(1-(4+a)^(1/2)))*(1+
(4+a)^(1/2))^(1/2)/(3+a-2*(-1+x)^2-(-1+x)^4)^(1/2)/((1+(-1+x)^2/(1-(4+a)^(1/2)))/(1+(-1+x)^2/(1+(4+a)^(1/2))))
^(1/2)

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Rubi [A]  time = 0.10, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1106, 1104, 418} \[ \frac {\sqrt {\sqrt {a+4}+1} \left (\frac {(x-1)^2}{1-\sqrt {a+4}}+1\right ) F\left (\tan ^{-1}\left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right )|-\frac {2 \sqrt {a+4}}{1-\sqrt {a+4}}\right )}{\sqrt {\frac {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}{\frac {(x-1)^2}{\sqrt {a+4}+1}+1}} \sqrt {a-(x-1)^4-2 (x-1)^2+3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 + Sqrt[4 + a]]*(1 + (-1 + x)^2/(1 - Sqrt[4 + a]))*EllipticF[ArcTan[(-1 + x)/Sqrt[1 + Sqrt[4 + a]]], (-
2*Sqrt[4 + a])/(1 - Sqrt[4 + a])])/(Sqrt[(1 + (-1 + x)^2/(1 - Sqrt[4 + a]))/(1 + (-1 + x)^2/(1 + Sqrt[4 + a]))
]*Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4])

Rule 418

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

Rule 1104

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

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

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Mathematica [B]  time = 1.49, size = 540, normalized size = 3.75 \[ \frac {2 \left (\sqrt {-\sqrt {a+4}-1}-x+1\right ) \sqrt {\frac {\sqrt {-\sqrt {a+4}-1} \left (\sqrt {\sqrt {a+4}-1}-x+1\right )}{\left (\sqrt {-\sqrt {a+4}-1}+\sqrt {\sqrt {a+4}-1}\right ) \left (\sqrt {-\sqrt {a+4}-1}-x+1\right )}} \left (\sqrt {-\sqrt {a+4}-1}+x-1\right ) \sqrt {\frac {\sqrt {-\sqrt {a+4}-1} \left (\sqrt {\sqrt {a+4}-1}+x-1\right )}{\left (\sqrt {\sqrt {a+4}-1}-\sqrt {-\sqrt {a+4}-1}\right ) \left (\sqrt {-\sqrt {a+4}-1}-x+1\right )}} F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {-\sqrt {a+4}-1}-\sqrt {\sqrt {a+4}-1}\right ) \left (x+\sqrt {-\sqrt {a+4}-1}-1\right )}{\left (\sqrt {-\sqrt {a+4}-1}+\sqrt {\sqrt {a+4}-1}\right ) \left (-x+\sqrt {-\sqrt {a+4}-1}+1\right )}}\right )|\frac {\left (\sqrt {-\sqrt {a+4}-1}+\sqrt {\sqrt {a+4}-1}\right )^2}{\left (\sqrt {-\sqrt {a+4}-1}-\sqrt {\sqrt {a+4}-1}\right )^2}\right )}{\sqrt {-\sqrt {a+4}-1} \sqrt {\frac {\left (\sqrt {-\sqrt {a+4}-1}-\sqrt {\sqrt {a+4}-1}\right ) \left (\sqrt {-\sqrt {a+4}-1}+x-1\right )}{\left (\sqrt {-\sqrt {a+4}-1}+\sqrt {\sqrt {a+4}-1}\right ) \left (\sqrt {-\sqrt {a+4}-1}-x+1\right )}} \sqrt {a-x \left (x^3-4 x^2+8 x-8\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(1 + Sqrt[-1 - Sqrt[4 + a]] - x)*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(1 + Sqrt[-1 + Sqrt[4 + a]] - x))/((Sqrt[-1 -
Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x)*S
qrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 + Sqrt[-1 + Sqrt[4 + a]] + x))/((-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 +
a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]*EllipticF[ArcSin[Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]
])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4
+ a]] - x))]], (Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])^2/(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4
+ a]])^2])/(Sqrt[-1 - Sqrt[4 + a]]*Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sq
rt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]*Sqrt[a
- x*(-8 + 8*x - 4*x^2 + x^3)])

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fricas [F]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x}}{x^{4} - 4 \, x^{3} + 8 \, x^{2} - a - 8 \, x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.03, size = 530, normalized size = 3.68 \[ -\frac {\left (\sqrt {-1-\sqrt {a +4}}+\sqrt {-1+\sqrt {a +4}}\right ) \sqrt {\frac {\left (-\sqrt {-1-\sqrt {a +4}}+\sqrt {-1+\sqrt {a +4}}\right ) \left (x -1-\sqrt {-1+\sqrt {a +4}}\right )}{\left (-\sqrt {-1-\sqrt {a +4}}-\sqrt {-1+\sqrt {a +4}}\right ) \left (x -1+\sqrt {-1+\sqrt {a +4}}\right )}}\, \left (x -1+\sqrt {-1+\sqrt {a +4}}\right )^{2} \sqrt {-\frac {2 \sqrt {-1+\sqrt {a +4}}\, \left (x -1-\sqrt {-1-\sqrt {a +4}}\right )}{\left (\sqrt {-1-\sqrt {a +4}}-\sqrt {-1+\sqrt {a +4}}\right ) \left (x -1+\sqrt {-1+\sqrt {a +4}}\right )}}\, \sqrt {-\frac {2 \sqrt {-1+\sqrt {a +4}}\, \left (x -1+\sqrt {-1-\sqrt {a +4}}\right )}{\left (-\sqrt {-1-\sqrt {a +4}}-\sqrt {-1+\sqrt {a +4}}\right ) \left (x -1+\sqrt {-1+\sqrt {a +4}}\right )}}\, \EllipticF \left (\sqrt {\frac {\left (-\sqrt {-1-\sqrt {a +4}}+\sqrt {-1+\sqrt {a +4}}\right ) \left (x -1-\sqrt {-1+\sqrt {a +4}}\right )}{\left (-\sqrt {-1-\sqrt {a +4}}-\sqrt {-1+\sqrt {a +4}}\right ) \left (x -1+\sqrt {-1+\sqrt {a +4}}\right )}}, \sqrt {\frac {\left (-\sqrt {-1-\sqrt {a +4}}-\sqrt {-1+\sqrt {a +4}}\right ) \left (\sqrt {-1-\sqrt {a +4}}+\sqrt {-1+\sqrt {a +4}}\right )}{\left (-\sqrt {-1-\sqrt {a +4}}+\sqrt {-1+\sqrt {a +4}}\right ) \left (\sqrt {-1-\sqrt {a +4}}-\sqrt {-1+\sqrt {a +4}}\right )}}\right )}{\left (-\sqrt {-1-\sqrt {a +4}}+\sqrt {-1+\sqrt {a +4}}\right ) \sqrt {-1+\sqrt {a +4}}\, \sqrt {-\left (x -1-\sqrt {-1+\sqrt {a +4}}\right ) \left (x -1+\sqrt {-1+\sqrt {a +4}}\right ) \left (x -1-\sqrt {-1-\sqrt {a +4}}\right ) \left (x -1+\sqrt {-1-\sqrt {a +4}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((-1-(a+4)^(1/2))^(1/2)+(-1+(a+4)^(1/2))^(1/2))*((-(-1-(a+4)^(1/2))^(1/2)+(-1+(a+4)^(1/2))^(1/2))*(x-1-(-1+(a
+4)^(1/2))^(1/2))/(-(-1-(a+4)^(1/2))^(1/2)-(-1+(a+4)^(1/2))^(1/2))/(x-1+(-1+(a+4)^(1/2))^(1/2)))^(1/2)*(x-1+(-
1+(a+4)^(1/2))^(1/2))^2*(-2*(-1+(a+4)^(1/2))^(1/2)*(x-1-(-1-(a+4)^(1/2))^(1/2))/((-1-(a+4)^(1/2))^(1/2)-(-1+(a
+4)^(1/2))^(1/2))/(x-1+(-1+(a+4)^(1/2))^(1/2)))^(1/2)*(-2*(-1+(a+4)^(1/2))^(1/2)*(x-1+(-1-(a+4)^(1/2))^(1/2))/
(-(-1-(a+4)^(1/2))^(1/2)-(-1+(a+4)^(1/2))^(1/2))/(x-1+(-1+(a+4)^(1/2))^(1/2)))^(1/2)/(-(-1-(a+4)^(1/2))^(1/2)+
(-1+(a+4)^(1/2))^(1/2))/(-1+(a+4)^(1/2))^(1/2)/(-(x-1-(-1+(a+4)^(1/2))^(1/2))*(x-1+(-1+(a+4)^(1/2))^(1/2))*(x-
1-(-1-(a+4)^(1/2))^(1/2))*(x-1+(-1-(a+4)^(1/2))^(1/2)))^(1/2)*EllipticF(((-(-1-(a+4)^(1/2))^(1/2)+(-1+(a+4)^(1
/2))^(1/2))*(x-1-(-1+(a+4)^(1/2))^(1/2))/(-(-1-(a+4)^(1/2))^(1/2)-(-1+(a+4)^(1/2))^(1/2))/(x-1+(-1+(a+4)^(1/2)
)^(1/2)))^(1/2),((-(-1-(a+4)^(1/2))^(1/2)-(-1+(a+4)^(1/2))^(1/2))*((-1-(a+4)^(1/2))^(1/2)+(-1+(a+4)^(1/2))^(1/
2))/(-(-1-(a+4)^(1/2))^(1/2)+(-1+(a+4)^(1/2))^(1/2))/((-1-(a+4)^(1/2))^(1/2)-(-1+(a+4)^(1/2))^(1/2)))^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {-x^4+4\,x^3-8\,x^2+8\,x+a}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(a + 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 \[ \int \frac {1}{\sqrt {a - x^{4} + 4 x^{3} - 8 x^{2} + 8 x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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