\(\int \frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{x} \, dx\) [2224]

Optimal result

Integrand size = 31, antiderivative size = 166 \[ \int \frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{x} \, dx=-8 \sqrt {1-\sqrt {1-\sqrt {\frac {-1+x}{x}}}}+2 \sqrt {-1+\sqrt {2}} \arctan \left (\frac {\sqrt {1-\sqrt {1-\sqrt {\frac {-1+x}{x}}}}}{\sqrt {-1+\sqrt {2}}}\right )+4 \text {arctanh}\left (\sqrt {1-\sqrt {1-\sqrt {\frac {-1+x}{x}}}}\right )+2 \sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {\frac {-1+x}{x}}}}}{\sqrt {1+\sqrt {2}}}\right ) \]

[Out]

-8*(1-(1-((-1+x)/x)^(1/2))^(1/2))^(1/2)+2*(2^(1/2)-1)^(1/2)*arctan((1-(1-((-1+x)/x)^(1/2))^(1/2))^(1/2)/(2^(1/
2)-1)^(1/2))+4*arctanh((1-(1-((-1+x)/x)^(1/2))^(1/2))^(1/2))+2*(1+2^(1/2))^(1/2)*arctanh((1-(1-((-1+x)/x)^(1/2
))^(1/2))^(1/2)/(1+2^(1/2))^(1/2))

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {6874, 2098, 213, 1180, 209} \[ \int \frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{x} \, dx=2 \sqrt {\sqrt {2}-1} \arctan \left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {\sqrt {2}-1}}\right )+4 \text {arctanh}\left (\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )+2 \sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {1+\sqrt {2}}}\right )-8 \sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \]

[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 209

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

Rule 213

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

Rule 1180

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 2098

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 6874

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

Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {\sqrt {1-\sqrt {1-x}} x}{1-x^2} \, dx,x,\sqrt {1-\frac {1}{x}}\right ) \\ & = -\left (4 \text {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 \text {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 \text {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 \text {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 \text {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 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )+4 \text {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 \text {arctanh}\left (\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )-\left (2 \left (1-\sqrt {2}\right )\right ) \text {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 ) \text {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}} \arctan \left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {-1+\sqrt {2}}}\right )+4 \text {arctanh}\left (\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )+2 \sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {1+\sqrt {2}}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{x} \, dx=-8 \sqrt {1-\sqrt {1-\sqrt {\frac {-1+x}{x}}}}+2 \sqrt {-1+\sqrt {2}} \arctan \left (\sqrt {1+\sqrt {2}} \sqrt {1-\sqrt {1-\sqrt {\frac {-1+x}{x}}}}\right )+4 \text {arctanh}\left (\sqrt {1-\sqrt {1-\sqrt {\frac {-1+x}{x}}}}\right )+2 \sqrt {1+\sqrt {2}} \text {arctanh}\left (\sqrt {-1+\sqrt {2}} \sqrt {1-\sqrt {1-\sqrt {\frac {-1+x}{x}}}}\right ) \]

[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[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]]]]

Maple [F]

\[\int \frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{x}d x\]

[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)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 253 vs. \(2 (126) = 252\).

Time = 0.25 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.52 \[ \int \frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{x} \, dx=\sqrt {\sqrt {2} + 1} \log \left (2 \, \sqrt {\sqrt {2} + 1} + 2 \, \sqrt {-\sqrt {-\sqrt {\frac {x - 1}{x}} + 1} + 1}\right ) - \sqrt {\sqrt {2} + 1} \log \left (-2 \, \sqrt {\sqrt {2} + 1} + 2 \, \sqrt {-\sqrt {-\sqrt {\frac {x - 1}{x}} + 1} + 1}\right ) + \frac {1}{2} \, \sqrt {-4 \, \sqrt {2} + 4} \log \left (\sqrt {-4 \, \sqrt {2} + 4} + 2 \, \sqrt {-\sqrt {-\sqrt {\frac {x - 1}{x}} + 1} + 1}\right ) - \frac {1}{2} \, \sqrt {-4 \, \sqrt {2} + 4} \log \left (-\sqrt {-4 \, \sqrt {2} + 4} + 2 \, \sqrt {-\sqrt {-\sqrt {\frac {x - 1}{x}} + 1} + 1}\right ) - 8 \, \sqrt {-\sqrt {-\sqrt {\frac {x - 1}{x}} + 1} + 1} + 2 \, \log \left (\sqrt {-\sqrt {-\sqrt {\frac {x - 1}{x}} + 1} + 1} + 1\right ) - 2 \, \log \left (\sqrt {-\sqrt {-\sqrt {\frac {x - 1}{x}} + 1} + 1} - 1\right ) \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{x} \, dx=\int \frac {\sqrt {1 - \sqrt {1 - \sqrt {1 - \frac {1}{x}}}}}{x}\, dx \]

[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)

Maxima [F]

\[ \int \frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{x} \, dx=\int { \frac {\sqrt {-\sqrt {-\sqrt {-\frac {1}{x} + 1} + 1} + 1}}{x} \,d x } \]

[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)

Giac [F]

\[ \int \frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{x} \, dx=\int { \frac {\sqrt {-\sqrt {-\sqrt {-\frac {1}{x} + 1} + 1} + 1}}{x} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{x} \, dx=\int \frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{x} \,d x \]

[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)