Integrand size = 31, antiderivative size = 617 \[ \int \frac {x^2}{\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}} \, dx=\frac {-4395 \sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}-148125 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{3/2}+184677 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{5/2}+598095 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{7/2}-1429914 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{9/2}+215274 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{11/2}+2531670 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{13/2}-3755230 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{15/2}+2613381 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{17/2}-1002573 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{19/2}+204165 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{21/2}-17265 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{23/2}}{30720 \left (-1-2 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )+\left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^2\right )^3 \left (1-\sqrt {1-\frac {1}{x}}\right )^3}+\left (-\frac {7}{32} \sqrt {\frac {1}{2} \left (\frac {1}{2}+\frac {1}{\sqrt {2}}\right )}-\frac {137 \sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}}}{1024}\right ) \arctan \left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {-1+\sqrt {2}}}\right )+\frac {703 \text {arctanh}\left (\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )}{2048}+\left (\frac {7}{32} \sqrt {\frac {1}{2} \left (-\frac {1}{2}+\frac {1}{\sqrt {2}}\right )}-\frac {137 \sqrt {-\frac {1}{2}+\frac {1}{\sqrt {2}}}}{1024}\right ) \text {arctanh}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {1+\sqrt {2}}}\right ) \]
1/30720*(-4395*(1-(1-(1-1/x)^(1/2))^(1/2))^(1/2)-148125*(1-(1-(1-1/x)^(1/2 ))^(1/2))^(3/2)+184677*(1-(1-(1-1/x)^(1/2))^(1/2))^(5/2)+598095*(1-(1-(1-1 /x)^(1/2))^(1/2))^(7/2)-1429914*(1-(1-(1-1/x)^(1/2))^(1/2))^(9/2)+215274*( 1-(1-(1-1/x)^(1/2))^(1/2))^(11/2)+2531670*(1-(1-(1-1/x)^(1/2))^(1/2))^(13/ 2)-3755230*(1-(1-(1-1/x)^(1/2))^(1/2))^(15/2)+2613381*(1-(1-(1-1/x)^(1/2)) ^(1/2))^(17/2)-1002573*(1-(1-(1-1/x)^(1/2))^(1/2))^(19/2)+204165*(1-(1-(1- 1/x)^(1/2))^(1/2))^(21/2)-17265*(1-(1-(1-1/x)^(1/2))^(1/2))^(23/2))/(-3+2* (1-(1-1/x)^(1/2))^(1/2)+(1-(1-(1-1/x)^(1/2))^(1/2))^2)^3/(1-(1-1/x)^(1/2)) ^3+(-7/64*(1+2^(1/2))^(1/2)-137/2048*(2+2*2^(1/2))^(1/2))*arctan((1-(1-(1- 1/x)^(1/2))^(1/2))^(1/2)/(2^(1/2)-1)^(1/2))+703/2048*arctanh((1-(1-(1-1/x) ^(1/2))^(1/2))^(1/2))+(7/64*(2^(1/2)-1)^(1/2)-137/2048*(-2+2*2^(1/2))^(1/2 ))*arctanh((1-(1-(1-1/x)^(1/2))^(1/2))^(1/2)/(1+2^(1/2))^(1/2))
Time = 2.22 (sec) , antiderivative size = 329, normalized size of antiderivative = 0.53 \[ \int \frac {x^2}{\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}} \, dx=\frac {\sqrt {1-\sqrt {1-\sqrt {\frac {-1+x}{x}}}} \sqrt {1-\sqrt {\frac {-1+x}{x}}} x \left (3177+17265 \sqrt {\frac {-1+x}{x}}+16 \left (57+773 \sqrt {\frac {-1+x}{x}}\right ) x+512 \left (1+21 \sqrt {\frac {-1+x}{x}}\right ) x^2\right )+2 \sqrt {1-\sqrt {1-\sqrt {\frac {-1+x}{x}}}} x \left (-87+7125 \sqrt {\frac {-1+x}{x}}+16 \left (-1+359 \sqrt {\frac {-1+x}{x}}\right ) x+5120 \sqrt {\frac {-1+x}{x}} x^2\right )-15 \sqrt {210466+149090 \sqrt {2}} \arctan \left (\sqrt {1+\sqrt {2}} \sqrt {1-\sqrt {1-\sqrt {\frac {-1+x}{x}}}}\right )+10545 \text {arctanh}\left (\sqrt {1-\sqrt {1-\sqrt {\frac {-1+x}{x}}}}\right )+94785 \sqrt {\frac {2}{105233+74545 \sqrt {2}}} \text {arctanh}\left (\sqrt {-1+\sqrt {2}} \sqrt {1-\sqrt {1-\sqrt {\frac {-1+x}{x}}}}\right )}{30720} \]
(Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x)/x]]]*Sqrt[1 - Sqrt[(-1 + x)/x]]*x*(3177 + 17265*Sqrt[(-1 + x)/x] + 16*(57 + 773*Sqrt[(-1 + x)/x])*x + 512*(1 + 21*S qrt[(-1 + x)/x])*x^2) + 2*Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x)/x]]]*x*(-87 + 71 25*Sqrt[(-1 + x)/x] + 16*(-1 + 359*Sqrt[(-1 + x)/x])*x + 5120*Sqrt[(-1 + x )/x]*x^2) - 15*Sqrt[210466 + 149090*Sqrt[2]]*ArcTan[Sqrt[1 + Sqrt[2]]*Sqrt [1 - Sqrt[1 - Sqrt[(-1 + x)/x]]]] + 10545*ArcTanh[Sqrt[1 - Sqrt[1 - Sqrt[( -1 + x)/x]]]] + 94785*Sqrt[2/(105233 + 74545*Sqrt[2])]*ArcTanh[Sqrt[-1 + S qrt[2]]*Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x)/x]]]])/30720
Time = 1.44 (sec) , antiderivative size = 980, normalized size of antiderivative = 1.59, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {7268, 7267, 2003, 2353, 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.
| \(\displaystyle \int \frac {x^2}{\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}} \, dx\) |
| \(\Big \downarrow \) 7268 |
| \(\displaystyle 2 \int \frac {\sqrt {1-\frac {1}{x}} x^4}{\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}d\sqrt {1-\frac {1}{x}}\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle -4 \int \frac {1}{\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (1-\frac {1}{x}\right )^{7/2} \left (1+\frac {1}{x}\right )^4 x}d\sqrt {1-\sqrt {1-\frac {1}{x}}}\) |
| \(\Big \downarrow \) 2003 |
| \(\displaystyle -4 \int \frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (\sqrt {1-\sqrt {1-\frac {1}{x}}}+1\right )}{\left (1-\frac {1}{x}\right )^{7/2} \left (1+\frac {1}{x}\right )^4}d\sqrt {1-\sqrt {1-\frac {1}{x}}}\) |
| \(\Big \downarrow \) 2353 |
| \(\displaystyle -4 \int \left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (-2 \sqrt {1-\sqrt {1-\frac {1}{x}}}-3\right )}{16 \left (-1-\frac {1}{x}\right )^3}+\frac {5 \sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{32 \sqrt {1-\frac {1}{x}}}-\frac {5 \sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (\sqrt {1-\sqrt {1-\frac {1}{x}}}+1\right )}{32 \left (-1-\frac {1}{x}\right )}+\frac {5 \sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{32 \left (1-\frac {1}{x}\right )}+\frac {5 \sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{32 \left (1-\frac {1}{x}\right )^{3/2}}+\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (5 \sqrt {1-\sqrt {1-\frac {1}{x}}}+6\right )}{32 \left (-1-\frac {1}{x}\right )^2}+\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{8 \left (1-\frac {1}{x}\right )^2}+\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{8 \left (1-\frac {1}{x}\right )^{5/2}}+\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{16 \left (1-\frac {1}{x}\right )^3}+\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{16 \left (1-\frac {1}{x}\right )^{7/2}}+\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (\sqrt {1-\sqrt {1-\frac {1}{x}}}+2\right )}{16 \left (-1-\frac {1}{x}\right )^4}\right )d\sqrt {1-\sqrt {1-\frac {1}{x}}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -4 \left (-\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (4-7 \sqrt {1-\sqrt {1-\frac {1}{x}}}\right )}{1536 \left (1+\frac {1}{x}\right )^2}-\frac {\sqrt {\frac {1}{2} \left (409+641 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {-1+\sqrt {2}}}\right )}{1024}-\frac {1}{128} \sqrt {\frac {1}{2} \left (7+73 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {-1+\sqrt {2}}}\right )+\frac {605 \arctan \left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {-1+\sqrt {2}}}\right )}{4096 \sqrt {2 \left (-1+\sqrt {2}\right )}}-\frac {703 \text {arctanh}\left (\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )}{8192}-\frac {\sqrt {\frac {1}{2} \left (-409+641 \sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {1+\sqrt {2}}}\right )}{1024}-\frac {1}{128} \sqrt {\frac {1}{2} \left (-7+73 \sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {1+\sqrt {2}}}\right )+\frac {605 \text {arctanh}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {1+\sqrt {2}}}\right )}{4096 \sqrt {2 \left (1+\sqrt {2}\right )}}-\frac {703 \sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{8192 \sqrt {1-\frac {1}{x}}}-\frac {703 \sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{12288 \left (1-\frac {1}{x}\right )}+\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (3 \sqrt {1-\sqrt {1-\frac {1}{x}}}+5\right )}{64 \left (1+\frac {1}{x}\right )}-\frac {\left (7-4 \sqrt {1-\sqrt {1-\frac {1}{x}}}\right ) \sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{512 \left (1+\frac {1}{x}\right )}-\frac {35 \sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{6144 \left (1+\frac {1}{x}\right )}-\frac {511 \sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{15360 \left (1-\frac {1}{x}\right )^{3/2}}-\frac {73 \sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{2560 \left (1-\frac {1}{x}\right )^2}+\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (3 \sqrt {1-\sqrt {1-\frac {1}{x}}}+4\right )}{128 \left (1+\frac {1}{x}\right )^2}-\frac {11 \sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{960 \left (1-\frac {1}{x}\right )^{5/2}}-\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{96 \left (1-\frac {1}{x}\right )^3}+\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (\sqrt {1-\sqrt {1-\frac {1}{x}}}+1\right )}{96 \left (1+\frac {1}{x}\right )^3}\right )\) |
-4*(-1/96*Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]]/(1 - x^(-1))^3 - (11*Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]])/(960*(1 - x^(-1))^(5/2)) - (73*Sqrt[1 - Sq rt[1 - Sqrt[1 - x^(-1)]]])/(2560*(1 - x^(-1))^2) - (511*Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]])/(15360*(1 - x^(-1))^(3/2)) - (703*Sqrt[1 - Sqrt[1 - Sq rt[1 - x^(-1)]]])/(12288*(1 - x^(-1))) - (703*Sqrt[1 - Sqrt[1 - Sqrt[1 - x ^(-1)]]])/(8192*Sqrt[1 - x^(-1)]) + (Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]]* (1 + Sqrt[1 - Sqrt[1 - x^(-1)]]))/(96*(1 + x^(-1))^3) - ((4 - 7*Sqrt[1 - S qrt[1 - x^(-1)]])*Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]])/(1536*(1 + x^(-1)) ^2) + (Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]]*(4 + 3*Sqrt[1 - Sqrt[1 - x^(-1 )]]))/(128*(1 + x^(-1))^2) - (35*Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]])/(61 44*(1 + x^(-1))) - ((7 - 4*Sqrt[1 - Sqrt[1 - x^(-1)]])*Sqrt[1 - Sqrt[1 - S qrt[1 - x^(-1)]]])/(512*(1 + x^(-1))) + (Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1) ]]]*(5 + 3*Sqrt[1 - Sqrt[1 - x^(-1)]]))/(64*(1 + x^(-1))) + (605*ArcTan[Sq rt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]]/Sqrt[-1 + Sqrt[2]]])/(4096*Sqrt[2*(-1 + Sqrt[2])]) - (Sqrt[(7 + 73*Sqrt[2])/2]*ArcTan[Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]]/Sqrt[-1 + Sqrt[2]]])/128 - (Sqrt[(409 + 641*Sqrt[2])/2]*ArcTan[S qrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]]/Sqrt[-1 + Sqrt[2]]])/1024 - (703*ArcTa nh[Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]]])/8192 + (605*ArcTanh[Sqrt[1 - Sqr t[1 - Sqrt[1 - x^(-1)]]]/Sqrt[1 + Sqrt[2]]])/(4096*Sqrt[2*(1 + Sqrt[2])]) - (Sqrt[(-7 + 73*Sqrt[2])/2]*ArcTanh[Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]...
3.32.9.3.1 Defintions of rubi rules used
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]))
Int[(Px_)*((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2) ^(p_), x_Symbol] :> Int[ExpandIntegrand[Px*(e*x)^m*(c + d*x)^n*(a + b*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && PolyQ[Px, x] && (Integer Q[p] || (IntegerQ[2*p] && IntegerQ[m] && ILtQ[n, 0]))
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]]
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]]
\[\int \frac {x^{2}}{\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}d x\]
Time = 0.26 (sec) , antiderivative size = 391, normalized size of antiderivative = 0.63 \[ \int \frac {x^2}{\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}} \, dx=\frac {1}{4096} \, \sqrt {2} \sqrt {74545 \, \sqrt {2} - 105233} \log \left (\sqrt {74545 \, \sqrt {2} - 105233} {\left (249 \, \sqrt {2} + 361\right )} + 6319 \, \sqrt {-\sqrt {-\sqrt {\frac {x - 1}{x}} + 1} + 1}\right ) - \frac {1}{4096} \, \sqrt {2} \sqrt {74545 \, \sqrt {2} - 105233} \log \left (-\sqrt {74545 \, \sqrt {2} - 105233} {\left (249 \, \sqrt {2} + 361\right )} + 6319 \, \sqrt {-\sqrt {-\sqrt {\frac {x - 1}{x}} + 1} + 1}\right ) + \frac {1}{4096} \, \sqrt {2} \sqrt {-74545 \, \sqrt {2} - 105233} \log \left ({\left (249 \, \sqrt {2} - 361\right )} \sqrt {-74545 \, \sqrt {2} - 105233} + 6319 \, \sqrt {-\sqrt {-\sqrt {\frac {x - 1}{x}} + 1} + 1}\right ) - \frac {1}{4096} \, \sqrt {2} \sqrt {-74545 \, \sqrt {2} - 105233} \log \left (-{\left (249 \, \sqrt {2} - 361\right )} \sqrt {-74545 \, \sqrt {2} - 105233} + 6319 \, \sqrt {-\sqrt {-\sqrt {\frac {x - 1}{x}} + 1} + 1}\right ) - \frac {1}{30720} \, {\left (32 \, x^{2} - {\left (512 \, x^{3} + 912 \, x^{2} + {\left (10752 \, x^{3} + 12368 \, x^{2} + 17265 \, x\right )} \sqrt {\frac {x - 1}{x}} + 3177 \, x\right )} \sqrt {-\sqrt {\frac {x - 1}{x}} + 1} - 2 \, {\left (5120 \, x^{3} + 5744 \, x^{2} + 7125 \, x\right )} \sqrt {\frac {x - 1}{x}} + 174 \, x\right )} \sqrt {-\sqrt {-\sqrt {\frac {x - 1}{x}} + 1} + 1} + \frac {703}{4096} \, \log \left (\sqrt {-\sqrt {-\sqrt {\frac {x - 1}{x}} + 1} + 1} + 1\right ) - \frac {703}{4096} \, \log \left (\sqrt {-\sqrt {-\sqrt {\frac {x - 1}{x}} + 1} + 1} - 1\right ) \]
1/4096*sqrt(2)*sqrt(74545*sqrt(2) - 105233)*log(sqrt(74545*sqrt(2) - 10523 3)*(249*sqrt(2) + 361) + 6319*sqrt(-sqrt(-sqrt((x - 1)/x) + 1) + 1)) - 1/4 096*sqrt(2)*sqrt(74545*sqrt(2) - 105233)*log(-sqrt(74545*sqrt(2) - 105233) *(249*sqrt(2) + 361) + 6319*sqrt(-sqrt(-sqrt((x - 1)/x) + 1) + 1)) + 1/409 6*sqrt(2)*sqrt(-74545*sqrt(2) - 105233)*log((249*sqrt(2) - 361)*sqrt(-7454 5*sqrt(2) - 105233) + 6319*sqrt(-sqrt(-sqrt((x - 1)/x) + 1) + 1)) - 1/4096 *sqrt(2)*sqrt(-74545*sqrt(2) - 105233)*log(-(249*sqrt(2) - 361)*sqrt(-7454 5*sqrt(2) - 105233) + 6319*sqrt(-sqrt(-sqrt((x - 1)/x) + 1) + 1)) - 1/3072 0*(32*x^2 - (512*x^3 + 912*x^2 + (10752*x^3 + 12368*x^2 + 17265*x)*sqrt((x - 1)/x) + 3177*x)*sqrt(-sqrt((x - 1)/x) + 1) - 2*(5120*x^3 + 5744*x^2 + 7 125*x)*sqrt((x - 1)/x) + 174*x)*sqrt(-sqrt(-sqrt((x - 1)/x) + 1) + 1) + 70 3/4096*log(sqrt(-sqrt(-sqrt((x - 1)/x) + 1) + 1) + 1) - 703/4096*log(sqrt( -sqrt(-sqrt((x - 1)/x) + 1) + 1) - 1)
\[ \int \frac {x^2}{\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}} \, dx=\int \frac {x^{2}}{\sqrt {1 - \sqrt {1 - \sqrt {1 - \frac {1}{x}}}}}\, dx \]
\[ \int \frac {x^2}{\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}} \, dx=\int { \frac {x^{2}}{\sqrt {-\sqrt {-\sqrt {-\frac {1}{x} + 1} + 1} + 1}} \,d x } \]
Timed out. \[ \int \frac {x^2}{\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {x^2}{\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}} \, dx=\int \frac {x^2}{\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}} \,d x \]