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

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]

(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])

_______________________________________________________________________________________

Rubi [A]  time = 0.237258, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{\sqrt{\sqrt{a+4}+1} \left (\frac{(1-x)^2}{1-\sqrt{a+4}}+1\right ) F\left (\tan ^{-1}\left (\frac{1-x}{\sqrt{\sqrt{a+4}+1}}\right )|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right )}{\sqrt{\frac{\frac{(1-x)^2}{1-\sqrt{a+4}}+1}{\frac{(1-x)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(1-x)^4-2 (1-x)^2+3}} \]

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]))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 27.1649, size = 112, normalized size = 0.78 \[ \frac{\left (\frac{\left (x - 1\right )^{2}}{- \sqrt{a + 4} + 1} + 1\right ) \sqrt{\sqrt{a + 4} + 1} F\left (\operatorname{atan}{\left (\frac{x - 1}{\sqrt{\sqrt{a + 4} + 1}} \right )}\middle | \frac{2 \sqrt{a + 4}}{\sqrt{a + 4} - 1}\right )}{\sqrt{\frac{- \frac{\left (x - 1\right )^{2}}{\sqrt{a + 4} - 1} + 1}{\frac{\left (x - 1\right )^{2}}{\sqrt{a + 4} + 1} + 1}} \sqrt{a - \left (x - 1\right )^{4} - 2 \left (x - 1\right )^{2} + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-x**4+4*x**3-8*x**2+a+8*x)**(1/2),x)

[Out]

((x - 1)**2/(-sqrt(a + 4) + 1) + 1)*sqrt(sqrt(a + 4) + 1)*elliptic_f(atan((x - 1
)/sqrt(sqrt(a + 4) + 1)), 2*sqrt(a + 4)/(sqrt(a + 4) - 1))/(sqrt((-(x - 1)**2/(s
qrt(a + 4) - 1) + 1)/((x - 1)**2/(sqrt(a + 4) + 1) + 1))*sqrt(a - (x - 1)**4 - 2
*(x - 1)**2 + 3))

_______________________________________________________________________________________

Mathematica [B]  time = 2.51937, 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)*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))]*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 - Sqrt[4 + a]] + x))/((Sqrt[-1 - S
qrt[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)])

_______________________________________________________________________________________

Maple [B]  time = 0.025, size = 530, normalized size = 3.7 \[ -{1 \left ( \sqrt{-1-\sqrt{4+a}}+\sqrt{-1+\sqrt{4+a}} \right ) \sqrt{{1 \left ( -\sqrt{-1-\sqrt{4+a}}+\sqrt{-1+\sqrt{4+a}} \right ) \left ( x-1-\sqrt{-1+\sqrt{4+a}} \right ) \left ( -\sqrt{-1-\sqrt{4+a}}-\sqrt{-1+\sqrt{4+a}} \right ) ^{-1} \left ( x-1+\sqrt{-1+\sqrt{4+a}} \right ) ^{-1}}} \left ( x-1+\sqrt{-1+\sqrt{4+a}} \right ) ^{2}\sqrt{-2\,{\frac{\sqrt{-1+\sqrt{4+a}} \left ( x-1-\sqrt{-1-\sqrt{4+a}} \right ) }{ \left ( \sqrt{-1-\sqrt{4+a}}-\sqrt{-1+\sqrt{4+a}} \right ) \left ( x-1+\sqrt{-1+\sqrt{4+a}} \right ) }}}\sqrt{-2\,{\frac{\sqrt{-1+\sqrt{4+a}} \left ( x-1+\sqrt{-1-\sqrt{4+a}} \right ) }{ \left ( -\sqrt{-1-\sqrt{4+a}}-\sqrt{-1+\sqrt{4+a}} \right ) \left ( x-1+\sqrt{-1+\sqrt{4+a}} \right ) }}}{\it EllipticF} \left ( \sqrt{{1 \left ( -\sqrt{-1-\sqrt{4+a}}+\sqrt{-1+\sqrt{4+a}} \right ) \left ( x-1-\sqrt{-1+\sqrt{4+a}} \right ) \left ( -\sqrt{-1-\sqrt{4+a}}-\sqrt{-1+\sqrt{4+a}} \right ) ^{-1} \left ( x-1+\sqrt{-1+\sqrt{4+a}} \right ) ^{-1}}},\sqrt{{1 \left ( -\sqrt{-1-\sqrt{4+a}}-\sqrt{-1+\sqrt{4+a}} \right ) \left ( \sqrt{-1-\sqrt{4+a}}+\sqrt{-1+\sqrt{4+a}} \right ) \left ( -\sqrt{-1-\sqrt{4+a}}+\sqrt{-1+\sqrt{4+a}} \right ) ^{-1} \left ( \sqrt{-1-\sqrt{4+a}}-\sqrt{-1+\sqrt{4+a}} \right ) ^{-1}}} \right ) \left ( -\sqrt{-1-\sqrt{4+a}}+\sqrt{-1+\sqrt{4+a}} \right ) ^{-1}{\frac{1}{\sqrt{-1+\sqrt{4+a}}}}{\frac{1}{\sqrt{- \left ( x-1-\sqrt{-1+\sqrt{4+a}} \right ) \left ( x-1+\sqrt{-1+\sqrt{4+a}} \right ) \left ( x-1-\sqrt{-1-\sqrt{4+a}} \right ) \left ( x-1+\sqrt{-1-\sqrt{4+a}} \right ) }}}} \]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

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)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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