3.26.38 \(\int \frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{x^2} \, dx\)

Optimal. Leaf size=213 \[ \sqrt {\frac {x-1}{x}} \left (\frac {8}{63} \sqrt {1-\sqrt {1-\sqrt {\frac {x-1}{x}}}} \sqrt {1-\sqrt {\frac {x-1}{x}}}-\frac {8}{315} \sqrt {1-\sqrt {1-\sqrt {\frac {x-1}{x}}}}\right )-\frac {64}{315} \sqrt {1-\sqrt {1-\sqrt {\frac {x-1}{x}}}}+\frac {64}{315} \sqrt {1-\sqrt {1-\sqrt {\frac {x-1}{x}}}} \sqrt {1-\sqrt {\frac {x-1}{x}}}+\frac {8 \sqrt {1-\sqrt {1-\sqrt {\frac {x-1}{x}}}} (x-1)}{9 x} \]

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Rubi [A]  time = 0.35, antiderivative size = 94, normalized size of antiderivative = 0.44, number of steps used = 6, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {6715, 371, 1398, 772} \begin {gather*} \frac {8}{9} \left (1-\sqrt {1-\sqrt {\frac {x-1}{x}}}\right )^{9/2}-\frac {24}{7} \left (1-\sqrt {1-\sqrt {\frac {x-1}{x}}}\right )^{7/2}+\frac {16}{5} \left (1-\sqrt {1-\sqrt {\frac {x-1}{x}}}\right )^{5/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]]/x^2,x]

[Out]

(16*(1 - Sqrt[1 - Sqrt[(-1 + x)/x]])^(5/2))/5 - (24*(1 - Sqrt[1 - Sqrt[(-1 + x)/x]])^(7/2))/7 + (8*(1 - Sqrt[1
- Sqrt[(-1 + x)/x]])^(9/2))/9

Rule 371

Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coefficient[v, x, 0], d = Coefficient[v,
x, 1]}, Dist[1/d^(m + 1), Subst[Int[SimplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]
] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]

Rule 772

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]

Rule 1398

Int[((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{g = Denominator[n]}, D
ist[g, Subst[Int[x^(g - 1)*(d + e*x^(g*n))^q*(a + c*x^(2*g*n))^p, x], x, x^(1/g)], x]] /; FreeQ[{a, c, d, e, p
, q}, x] && EqQ[n2, 2*n] && FractionQ[n]

Rule 6715

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rubi steps

\begin {align*} \int \frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{x^2} \, dx &=-\operatorname {Subst}\left (\int \sqrt {1-\sqrt {1-\sqrt {1-x}}} \, dx,x,\frac {1}{x}\right )\\ &=2 \operatorname {Subst}\left (\int \sqrt {1-\sqrt {1-x}} x \, dx,x,\sqrt {\frac {-1+x}{x}}\right )\\ &=2 \operatorname {Subst}\left (\int \sqrt {1-\sqrt {x}} (-1+x) \, dx,x,1-\sqrt {\frac {-1+x}{x}}\right )\\ &=4 \operatorname {Subst}\left (\int \sqrt {1-x} x \left (-1+x^2\right ) \, dx,x,\sqrt {1-\sqrt {\frac {-1+x}{x}}}\right )\\ &=4 \operatorname {Subst}\left (\int \left (-2 (1-x)^{3/2}+3 (1-x)^{5/2}-(1-x)^{7/2}\right ) \, dx,x,\sqrt {1-\sqrt {\frac {-1+x}{x}}}\right )\\ &=\frac {16}{5} \left (1-\sqrt {1-\sqrt {-\frac {1-x}{x}}}\right )^{5/2}-\frac {24}{7} \left (1-\sqrt {1-\sqrt {-\frac {1-x}{x}}}\right )^{7/2}+\frac {8}{9} \left (1-\sqrt {1-\sqrt {-\frac {1-x}{x}}}\right )^{9/2}\\ \end {align*}

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Mathematica [A]  time = 0.14, size = 67, normalized size = 0.31 \begin {gather*} \frac {8}{315} \left (1-\sqrt {1-\sqrt {\frac {x-1}{x}}}\right )^{5/2} \left (65 \sqrt {1-\sqrt {\frac {x-1}{x}}}-35 \sqrt {\frac {x-1}{x}}+61\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]]/x^2,x]

[Out]

(8*(1 - Sqrt[1 - Sqrt[(-1 + x)/x]])^(5/2)*(61 + 65*Sqrt[1 - Sqrt[(-1 + x)/x]] - 35*Sqrt[(-1 + x)/x]))/315

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IntegrateAlgebraic [A]  time = 2.20, size = 184, normalized size = 0.86 \begin {gather*} \frac {64}{315} \sqrt {1-\sqrt {1-\sqrt {\frac {-1+x}{x}}}} \sqrt {1-\sqrt {\frac {-1+x}{x}}}+\left (-\frac {8}{315} \sqrt {1-\sqrt {1-\sqrt {\frac {-1+x}{x}}}}+\frac {8}{63} \sqrt {1-\sqrt {1-\sqrt {\frac {-1+x}{x}}}} \sqrt {1-\sqrt {\frac {-1+x}{x}}}\right ) \sqrt {\frac {-1+x}{x}}+\frac {8 \sqrt {1-\sqrt {1-\sqrt {\frac {-1+x}{x}}}} (-35+27 x)}{315 x} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]]/x^2,x]

[Out]

(64*Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x)/x]]]*Sqrt[1 - Sqrt[(-1 + x)/x]])/315 + ((-8*Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x
)/x]]])/315 + (8*Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x)/x]]]*Sqrt[1 - Sqrt[(-1 + x)/x]])/63)*Sqrt[(-1 + x)/x] + (8*Sq
rt[1 - Sqrt[1 - Sqrt[(-1 + x)/x]]]*(-35 + 27*x))/(315*x)

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fricas [A]  time = 0.93, size = 75, normalized size = 0.35 \begin {gather*} \frac {8 \, {\left ({\left (5 \, x \sqrt {\frac {x - 1}{x}} + 8 \, x\right )} \sqrt {-\sqrt {\frac {x - 1}{x}} + 1} - x \sqrt {\frac {x - 1}{x}} + 27 \, x - 35\right )} \sqrt {-\sqrt {-\sqrt {\frac {x - 1}{x}} + 1} + 1}}{315 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-(1-(1-1/x)^(1/2))^(1/2))^(1/2)/x^2,x, algorithm="fricas")

[Out]

8/315*((5*x*sqrt((x - 1)/x) + 8*x)*sqrt(-sqrt((x - 1)/x) + 1) - x*sqrt((x - 1)/x) + 27*x - 35)*sqrt(-sqrt(-sqr
t((x - 1)/x) + 1) + 1)/x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {-\sqrt {-\sqrt {-\frac {1}{x} + 1} + 1} + 1}}{x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-(1-(1-1/x)^(1/2))^(1/2))^(1/2)/x^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.67, size = 71, normalized size = 0.33

method	result	size
derivativedivides	\(\frac {8 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{\frac {9}{2}}}{9}-\frac {24 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{\frac {7}{2}}}{7}+\frac {16 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{\frac {5}{2}}}{5}\)	\(71\)
default	\(\frac {8 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{\frac {9}{2}}}{9}-\frac {24 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{\frac {7}{2}}}{7}+\frac {16 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{\frac {5}{2}}}{5}\)	\(71\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-(1-(1-1/x)^(1/2))^(1/2))^(1/2)/x^2,x,method=_RETURNVERBOSE)

[Out]

8/9*(1-(1-(1-1/x)^(1/2))^(1/2))^(9/2)-24/7*(1-(1-(1-1/x)^(1/2))^(1/2))^(7/2)+16/5*(1-(1-(1-1/x)^(1/2))^(1/2))^
(5/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {-\sqrt {-\sqrt {-\frac {1}{x} + 1} + 1} + 1}}{x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-(1-(1-1/x)^(1/2))^(1/2))^(1/2)/x^2,x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - (1 - (1 - 1/x)^(1/2))^(1/2))^(1/2)/x^2,x)

[Out]

int((1 - (1 - (1 - 1/x)^(1/2))^(1/2))^(1/2)/x^2, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {1 - \sqrt {1 - \sqrt {1 - \frac {1}{x}}}}}{x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-(1-(1-1/x)**(1/2))**(1/2))**(1/2)/x**2,x)

[Out]

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

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