3.23.24 \(\int \frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{x} \, dx\)

Optimal. Leaf size=166 \[ -8 \sqrt {1-\sqrt {1-\sqrt {\frac {x-1}{x}}}}+2 \sqrt {\sqrt {2}-1} \tan ^{-1}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {\frac {x-1}{x}}}}}{\sqrt {\sqrt {2}-1}}\right )+4 \tanh ^{-1}\left (\sqrt {1-\sqrt {1-\sqrt {\frac {x-1}{x}}}}\right )+2 \sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {\frac {x-1}{x}}}}}{\sqrt {1+\sqrt {2}}}\right ) \]

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Rubi [A]  time = 0.98, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {6742, 2073, 207, 1166, 203} \begin {gather*} -8 \sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}+2 \sqrt {\sqrt {2}-1} \tan ^{-1}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {\sqrt {2}-1}}\right )+4 \tanh ^{-1}\left (\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )+2 \sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {1+\sqrt {2}}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-8*Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]] + 2*Sqrt[-1 + Sqrt[2]]*ArcTan[Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]]/Sqr
t[-1 + Sqrt[2]]] + 4*ArcTanh[Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]]] + 2*Sqrt[1 + Sqrt[2]]*ArcTanh[Sqrt[1 - Sqrt
[1 - Sqrt[1 - x^(-1)]]]/Sqrt[1 + Sqrt[2]]]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 1166

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

Rule 2073

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P /. x -> Sqrt[x]]}, Int[ExpandIntegrand[(PP /. x ->
x^2)^p*Q^q, x], x] /;  !SumQ[NonfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x^2] && PolyQ[Q, x] && ILtQ[p,
0]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

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

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Mathematica [A]  time = 0.26, size = 166, normalized size = 1.00 \begin {gather*} -8 \sqrt {1-\sqrt {1-\sqrt {\frac {x-1}{x}}}}+2 \sqrt {\sqrt {2}-1} \tan ^{-1}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {\frac {x-1}{x}}}}}{\sqrt {\sqrt {2}-1}}\right )+4 \tanh ^{-1}\left (\sqrt {1-\sqrt {1-\sqrt {\frac {x-1}{x}}}}\right )+2 \sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {\frac {x-1}{x}}}}}{\sqrt {1+\sqrt {2}}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-8*Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x)/x]]] + 2*Sqrt[-1 + Sqrt[2]]*ArcTan[Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x)/x]]]/Sqr
t[-1 + Sqrt[2]]] + 4*ArcTanh[Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x)/x]]]] + 2*Sqrt[1 + Sqrt[2]]*ArcTanh[Sqrt[1 - Sqrt
[1 - Sqrt[(-1 + x)/x]]]/Sqrt[1 + Sqrt[2]]]

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IntegrateAlgebraic [A]  time = 1.26, size = 166, normalized size = 1.00 \begin {gather*} -8 \sqrt {1-\sqrt {1-\sqrt {\frac {-1+x}{x}}}}+2 \sqrt {-1+\sqrt {2}} \tan ^{-1}\left (\sqrt {1+\sqrt {2}} \sqrt {1-\sqrt {1-\sqrt {\frac {-1+x}{x}}}}\right )+4 \tanh ^{-1}\left (\sqrt {1-\sqrt {1-\sqrt {\frac {-1+x}{x}}}}\right )+2 \sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\sqrt {-1+\sqrt {2}} \sqrt {1-\sqrt {1-\sqrt {\frac {-1+x}{x}}}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-8*Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x)/x]]] + 2*Sqrt[-1 + Sqrt[2]]*ArcTan[Sqrt[1 + Sqrt[2]]*Sqrt[1 - Sqrt[1 - Sqrt
[(-1 + x)/x]]]] + 4*ArcTanh[Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x)/x]]]] + 2*Sqrt[1 + Sqrt[2]]*ArcTanh[Sqrt[-1 + Sqrt
[2]]*Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x)/x]]]]

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fricas [A]  time = 0.70, size = 243, normalized size = 1.46 \begin {gather*} 4 \, \sqrt {\sqrt {2} - 1} \arctan \left (-\sqrt {\sqrt {2} - \sqrt {-\sqrt {\frac {x - 1}{x}} + 1}} {\left (\sqrt {2} + 1\right )} \sqrt {\sqrt {2} - 1} + {\left (\sqrt {2} + 1\right )} \sqrt {\sqrt {2} - 1} \sqrt {-\sqrt {-\sqrt {\frac {x - 1}{x}} + 1} + 1}\right ) + \sqrt {\sqrt {2} + 1} \log \left (2 \, \sqrt {\sqrt {2} + 1} + 2 \, \sqrt {-\sqrt {-\sqrt {\frac {x - 1}{x}} + 1} + 1}\right ) - \sqrt {\sqrt {2} + 1} \log \left (-2 \, \sqrt {\sqrt {2} + 1} + 2 \, \sqrt {-\sqrt {-\sqrt {\frac {x - 1}{x}} + 1} + 1}\right ) - 8 \, \sqrt {-\sqrt {-\sqrt {\frac {x - 1}{x}} + 1} + 1} + 2 \, \log \left (\sqrt {-\sqrt {-\sqrt {\frac {x - 1}{x}} + 1} + 1} + 1\right ) - 2 \, \log \left (\sqrt {-\sqrt {-\sqrt {\frac {x - 1}{x}} + 1} + 1} - 1\right ) \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,x, algorithm="fricas")

[Out]

4*sqrt(sqrt(2) - 1)*arctan(-sqrt(sqrt(2) - sqrt(-sqrt((x - 1)/x) + 1))*(sqrt(2) + 1)*sqrt(sqrt(2) - 1) + (sqrt
(2) + 1)*sqrt(sqrt(2) - 1)*sqrt(-sqrt(-sqrt((x - 1)/x) + 1) + 1)) + sqrt(sqrt(2) + 1)*log(2*sqrt(sqrt(2) + 1)
+ 2*sqrt(-sqrt(-sqrt((x - 1)/x) + 1) + 1)) - sqrt(sqrt(2) + 1)*log(-2*sqrt(sqrt(2) + 1) + 2*sqrt(-sqrt(-sqrt((
x - 1)/x) + 1) + 1)) - 8*sqrt(-sqrt(-sqrt((x - 1)/x) + 1) + 1) + 2*log(sqrt(-sqrt(-sqrt((x - 1)/x) + 1) + 1) +
1) - 2*log(sqrt(-sqrt(-sqrt((x - 1)/x) + 1) + 1) - 1)

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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}\,{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,x, algorithm="giac")

[Out]

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

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maple [F]  time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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}\,{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,x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{x} \,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,x)

[Out]

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

[Out]

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

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