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

3.23.24.1 Optimal result

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

output

-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))
 

3.23.24.2 Mathematica [A] (verified)

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

input

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

output

-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 - S 
qrt[1 - Sqrt[(-1 + x)/x]]]] + 2*Sqrt[1 + Sqrt[2]]*ArcTanh[Sqrt[-1 + Sqrt[2 
]]*Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x)/x]]]]
 

3.23.24.3 Rubi [A] (verified)

Time = 1.10 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.88, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.452, Rules used = {7268, 7267, 2003, 2351, 481, 25, 561, 654, 25, 1480, 217, 219, 1610, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 7268

\(\displaystyle 2 \int \sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \sqrt {1-\frac {1}{x}} xd\sqrt {1-\frac {1}{x}}\)

\(\Big \downarrow \) 7267

\(\displaystyle -4 \int \frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {1-\frac {1}{x}} \left (1+\frac {1}{x}\right ) x}d\sqrt {1-\sqrt {1-\frac {1}{x}}}\)

\(\Big \downarrow \) 2003

\(\displaystyle -4 \int \frac {\left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{3/2} \left (\sqrt {1-\sqrt {1-\frac {1}{x}}}+1\right )}{\sqrt {1-\frac {1}{x}} \left (1+\frac {1}{x}\right )}d\sqrt {1-\sqrt {1-\frac {1}{x}}}\)

\(\Big \downarrow \) 2351

\(\displaystyle -4 \left (\int \frac {\left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{3/2}}{1+\frac {1}{x}}d\sqrt {1-\sqrt {1-\frac {1}{x}}}+\int \frac {\left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{3/2}}{\sqrt {1-\frac {1}{x}} \left (1+\frac {1}{x}\right )}d\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )\)

\(\Big \downarrow \) 481

\(\displaystyle -4 \left (-\int -\frac {3-2 \sqrt {1-\sqrt {1-\frac {1}{x}}}}{\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (1+\frac {1}{x}\right )}d\sqrt {1-\sqrt {1-\frac {1}{x}}}+\int \frac {\left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{3/2}}{\sqrt {1-\frac {1}{x}} \left (1+\frac {1}{x}\right )}d\sqrt {1-\sqrt {1-\frac {1}{x}}}+2 \sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle -4 \left (\int \frac {3-2 \sqrt {1-\sqrt {1-\frac {1}{x}}}}{\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (1+\frac {1}{x}\right )}d\sqrt {1-\sqrt {1-\frac {1}{x}}}+\int \frac {\left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{3/2}}{\sqrt {1-\frac {1}{x}} \left (1+\frac {1}{x}\right )}d\sqrt {1-\sqrt {1-\frac {1}{x}}}+2 \sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )\)

\(\Big \downarrow \) 561

\(\displaystyle -4 \left (\int \frac {3-2 \sqrt {1-\sqrt {1-\frac {1}{x}}}}{\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (1+\frac {1}{x}\right )}d\sqrt {1-\sqrt {1-\frac {1}{x}}}-2 \int \frac {\left (1-\frac {1}{x}\right )^2 x}{-\left (1-\frac {1}{x}\right )^2+2 \left (1-\frac {1}{x}\right )+1}d\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}+2 \sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )\)

\(\Big \downarrow \) 654

\(\displaystyle -4 \left (2 \int -\frac {2 \left (1-\frac {1}{x}\right )+1}{-\left (1-\frac {1}{x}\right )^2+2 \left (1-\frac {1}{x}\right )+1}d\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}-2 \int \frac {\left (1-\frac {1}{x}\right )^2 x}{-\left (1-\frac {1}{x}\right )^2+2 \left (1-\frac {1}{x}\right )+1}d\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}+2 \sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle -4 \left (-2 \int \frac {2 \left (1-\frac {1}{x}\right )+1}{-\left (1-\frac {1}{x}\right )^2+2 \left (1-\frac {1}{x}\right )+1}d\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}-2 \int \frac {\left (1-\frac {1}{x}\right )^2 x}{-\left (1-\frac {1}{x}\right )^2+2 \left (1-\frac {1}{x}\right )+1}d\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}+2 \sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )\)

\(\Big \downarrow \) 1480

\(\displaystyle -4 \left (2 \left (-\frac {1}{4} \left (4-3 \sqrt {2}\right ) \int \frac {1}{\frac {1}{x}-\sqrt {2}}d\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}-\frac {1}{4} \left (4+3 \sqrt {2}\right ) \int \frac {1}{\sqrt {2}+\frac {1}{x}}d\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )-2 \int \frac {\left (1-\frac {1}{x}\right )^2 x}{-\left (1-\frac {1}{x}\right )^2+2 \left (1-\frac {1}{x}\right )+1}d\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}+2 \sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )\)

\(\Big \downarrow \) 217

\(\displaystyle -4 \left (2 \left (\frac {\left (4-3 \sqrt {2}\right ) \arctan \left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {\sqrt {2}-1}}\right )}{4 \sqrt {\sqrt {2}-1}}-\frac {1}{4} \left (4+3 \sqrt {2}\right ) \int \frac {1}{\sqrt {2}+\frac {1}{x}}d\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )-2 \int \frac {\left (1-\frac {1}{x}\right )^2 x}{-\left (1-\frac {1}{x}\right )^2+2 \left (1-\frac {1}{x}\right )+1}d\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}+2 \sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle -4 \left (-2 \int \frac {\left (1-\frac {1}{x}\right )^2 x}{-\left (1-\frac {1}{x}\right )^2+2 \left (1-\frac {1}{x}\right )+1}d\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}+2 \left (\frac {\left (4-3 \sqrt {2}\right ) \arctan \left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {\sqrt {2}-1}}\right )}{4 \sqrt {\sqrt {2}-1}}-\frac {\left (4+3 \sqrt {2}\right ) \text {arctanh}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {1+\sqrt {2}}}\right )}{4 \sqrt {1+\sqrt {2}}}\right )+2 \sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )\)

\(\Big \downarrow \) 1610

\(\displaystyle -4 \left (-2 \int \left (\frac {3 \left (1-\frac {1}{x}\right )+1}{2 \left (\left (1-\frac {1}{x}\right )^2-2 \left (1-\frac {1}{x}\right )-1\right )}+\frac {x}{2}\right )d\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}+2 \left (\frac {\left (4-3 \sqrt {2}\right ) \arctan \left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {\sqrt {2}-1}}\right )}{4 \sqrt {\sqrt {2}-1}}-\frac {\left (4+3 \sqrt {2}\right ) \text {arctanh}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {1+\sqrt {2}}}\right )}{4 \sqrt {1+\sqrt {2}}}\right )+2 \sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle -4 \left (2 \left (\frac {\left (4-3 \sqrt {2}\right ) \arctan \left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {\sqrt {2}-1}}\right )}{4 \sqrt {\sqrt {2}-1}}-\frac {\left (4+3 \sqrt {2}\right ) \text {arctanh}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {1+\sqrt {2}}}\right )}{4 \sqrt {1+\sqrt {2}}}\right )-2 \left (\frac {1}{4} \sqrt {5 \sqrt {2}-7} \arctan \left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {\sqrt {2}-1}}\right )+\frac {1}{2} \text {arctanh}\left (\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )-\frac {1}{4} \sqrt {7+5 \sqrt {2}} \text {arctanh}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {1+\sqrt {2}}}\right )\right )+2 \sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )\)

input

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

output

-4*(2*Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]] + 2*(((4 - 3*Sqrt[2])*ArcTan[Sq 
rt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]]/Sqrt[-1 + Sqrt[2]]])/(4*Sqrt[-1 + Sqrt[ 
2]]) - ((4 + 3*Sqrt[2])*ArcTanh[Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]]/Sqrt[ 
1 + Sqrt[2]]])/(4*Sqrt[1 + Sqrt[2]])) - 2*((Sqrt[-7 + 5*Sqrt[2]]*ArcTan[Sq 
rt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]]/Sqrt[-1 + Sqrt[2]]])/4 + ArcTanh[Sqrt[1 
 - Sqrt[1 - Sqrt[1 - x^(-1)]]]]/2 - (Sqrt[7 + 5*Sqrt[2]]*ArcTanh[Sqrt[1 - 
Sqrt[1 - Sqrt[1 - x^(-1)]]]/Sqrt[1 + Sqrt[2]]])/4))
 

3.23.24.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 217

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

rule 219

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

rule 481

Int[((c_) + (d_.)*(x_))^(n_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[d*((c 
 + d*x)^(n - 1)/(b*(n - 1))), x] + Simp[1/b   Int[(c + d*x)^(n - 2)*(Simp[b 
*c^2 - a*d^2 + 2*b*c*d*x, x]/(a + b*x^2)), x], x] /; FreeQ[{a, b, c, d}, x] 
 && GtQ[n, 1]
 

rule 561

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> With[{k = Denominator[n]}, Simp[k/d   Subst[Int[x^(k*(n + 1) - 1)*(-c 
/d + x^k/d)^m*Simp[(b*c^2 + a*d^2)/d^2 - 2*b*c*(x^k/d^2) + b*(x^(2*k)/d^2), 
 x]^p, x], x, (c + d*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, m, p}, x] && Frac 
tionQ[n] && IntegerQ[p] && IntegerQ[m]
 

rule 654

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), 
x_Symbol] :> Simp[2   Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d* 
x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
 

rule 1480

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q))   Int[1/( 
b/2 - q/2 + c*x^2), x], x] + Simp[(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 1610

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.))/((a_) + (b_.)*(x_)^2 + 
 (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*((d + e*x^2)^q/(a 
+ b*x^2 + c*x^4)), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^2 - 4 
*a*c, 0] && IntegerQ[q] && IntegerQ[m]
 

rule 2003

Int[(u_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] : 
> Int[u*(c + d*x)^(n + p)*(a/c + (b/d)*x)^p, x] /; FreeQ[{a, b, c, d, n, p} 
, x] && EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[c, 0] && 
  !IntegerQ[n]))
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2351

Int[((Px_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_.))/(x_), x_S 
ymbol] :> Int[PolynomialQuotient[Px, x, x]*(c + d*x)^n*(a + b*x^2)^p, x] + 
Simp[PolynomialRemainder[Px, x, x]   Int[(c + d*x)^n*((a + b*x^2)^p/x), x], 
 x] /; FreeQ[{a, b, c, d, n, p}, x] && PolynomialQ[Px, x]
 

rule 7267

Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si 
mp[lst[[2]]*lst[[4]]   Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x 
] /;  !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
 

rule 7268

Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfQuotientOfLinears 
[u, x]}, Simp[lst[[2]]*lst[[4]]   Subst[Int[lst[[1]], x], x, lst[[3]]^(1/ls 
t[[2]])], x] /;  !FalseQ[lst]]
 

3.23.24.4 Maple [F]

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

input

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

output

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

3.23.24.5 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 \[ \int \frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{x} \, dx=\sqrt {\sqrt {2} + 1} \log \left (2 \, \sqrt {\sqrt {2} + 1} + 2 \, \sqrt {-\sqrt {-\sqrt {\frac {x - 1}{x}} + 1} + 1}\right ) - \sqrt {\sqrt {2} + 1} \log \left (-2 \, \sqrt {\sqrt {2} + 1} + 2 \, \sqrt {-\sqrt {-\sqrt {\frac {x - 1}{x}} + 1} + 1}\right ) + \frac {1}{2} \, \sqrt {-4 \, \sqrt {2} + 4} \log \left (\sqrt {-4 \, \sqrt {2} + 4} + 2 \, \sqrt {-\sqrt {-\sqrt {\frac {x - 1}{x}} + 1} + 1}\right ) - \frac {1}{2} \, \sqrt {-4 \, \sqrt {2} + 4} \log \left (-\sqrt {-4 \, \sqrt {2} + 4} + 2 \, \sqrt {-\sqrt {-\sqrt {\frac {x - 1}{x}} + 1} + 1}\right ) - 8 \, \sqrt {-\sqrt {-\sqrt {\frac {x - 1}{x}} + 1} + 1} + 2 \, \log \left (\sqrt {-\sqrt {-\sqrt {\frac {x - 1}{x}} + 1} + 1} + 1\right ) - 2 \, \log \left (\sqrt {-\sqrt {-\sqrt {\frac {x - 1}{x}} + 1} + 1} - 1\right ) \]

input

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

output

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(-sq 
rt((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)*l 
og(-sqrt(-4*sqrt(2) + 4) + 2*sqrt(-sqrt(-sqrt((x - 1)/x) + 1) + 1)) - 8*sq 
rt(-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)
 

3.23.24.6 Sympy [F]

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

input

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

output

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

3.23.24.7 Maxima [F]

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

input

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

output

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

3.23.24.8 Giac [F]

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

input

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

output

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

3.23.24.9 Mupad [F(-1)]

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

input

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

output

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