Integrand size = 29, antiderivative size = 501 \[ \int \frac {x}{\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}} \, dx=\frac {69 \sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}+1787 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{3/2}-2139 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{5/2}-3301 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{7/2}+8403 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{9/2}-6611 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{11/2}+2275 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{13/2}-291 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{15/2}}{384 \left (-1-2 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )+\left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^2\right )^2 \left (1-\sqrt {1-\frac {1}{x}}\right )^2}+\left (-\frac {19}{64} \sqrt {\frac {1}{2} \left (\frac {1}{2}+\frac {1}{\sqrt {2}}\right )}-\frac {11}{64} \sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}}\right ) \arctan \left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {-1+\sqrt {2}}}\right )+\frac {59}{128} \text {arctanh}\left (\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )+\left (\frac {19}{64} \sqrt {\frac {1}{2} \left (-\frac {1}{2}+\frac {1}{\sqrt {2}}\right )}-\frac {11}{64} \sqrt {-\frac {1}{2}+\frac {1}{\sqrt {2}}}\right ) \text {arctanh}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {1+\sqrt {2}}}\right ) \]
1/384*(69*(1-(1-(1-1/x)^(1/2))^(1/2))^(1/2)+1787*(1-(1-(1-1/x)^(1/2))^(1/2 ))^(3/2)-2139*(1-(1-(1-1/x)^(1/2))^(1/2))^(5/2)-3301*(1-(1-(1-1/x)^(1/2))^ (1/2))^(7/2)+8403*(1-(1-(1-1/x)^(1/2))^(1/2))^(9/2)-6611*(1-(1-(1-1/x)^(1/ 2))^(1/2))^(11/2)+2275*(1-(1-(1-1/x)^(1/2))^(1/2))^(13/2)-291*(1-(1-(1-1/x )^(1/2))^(1/2))^(15/2))/(-3+2*(1-(1-1/x)^(1/2))^(1/2)+(1-(1-(1-1/x)^(1/2)) ^(1/2))^2)^2/(1-(1-1/x)^(1/2))^2+(-19/128*(1+2^(1/2))^(1/2)-11/128*(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))+ 59/128*arctanh((1-(1-(1-1/x)^(1/2))^(1/2))^(1/2))+(19/128*(2^(1/2)-1)^(1/2 )-11/128*(-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 = 1.44 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.57 \[ \int \frac {x}{\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}} \, dx=\frac {1}{384} \left (\sqrt {1-\sqrt {1-\sqrt {\frac {-1+x}{x}}}} \sqrt {1-\sqrt {\frac {-1+x}{x}}} x \left (55+291 \sqrt {\frac {-1+x}{x}}+16 \left (1+13 \sqrt {\frac {-1+x}{x}}\right ) x\right )+2 \sqrt {1-\sqrt {1-\sqrt {\frac {-1+x}{x}}}} x \left (-1+119 \sqrt {\frac {-1+x}{x}}+96 \sqrt {\frac {-1+x}{x}} x\right )-3 \sqrt {1439+1021 \sqrt {2}} \arctan \left (\sqrt {1+\sqrt {2}} \sqrt {1-\sqrt {1-\sqrt {\frac {-1+x}{x}}}}\right )+177 \text {arctanh}\left (\sqrt {1-\sqrt {1-\sqrt {\frac {-1+x}{x}}}}\right )+\frac {357 \text {arctanh}\left (\sqrt {-1+\sqrt {2}} \sqrt {1-\sqrt {1-\sqrt {\frac {-1+x}{x}}}}\right )}{\sqrt {1439+1021 \sqrt {2}}}\right ) \]
(Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x)/x]]]*Sqrt[1 - Sqrt[(-1 + x)/x]]*x*(55 + 2 91*Sqrt[(-1 + x)/x] + 16*(1 + 13*Sqrt[(-1 + x)/x])*x) + 2*Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x)/x]]]*x*(-1 + 119*Sqrt[(-1 + x)/x] + 96*Sqrt[(-1 + x)/x]*x) - 3*Sqrt[1439 + 1021*Sqrt[2]]*ArcTan[Sqrt[1 + Sqrt[2]]*Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x)/x]]]] + 177*ArcTanh[Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x)/x]]]] + (357*ArcTanh[Sqrt[-1 + Sqrt[2]]*Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x)/x]]]])/Sqr t[1439 + 1021*Sqrt[2]])/384
Time = 1.25 (sec) , antiderivative size = 730, normalized size of antiderivative = 1.46, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {7268, 25, 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.
| \(\displaystyle \int \frac {x}{\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}} \, dx\) |
| \(\Big \downarrow \) 7268 |
| \(\displaystyle -2 \int -\frac {\sqrt {1-\frac {1}{x}} x^3}{\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}d\sqrt {1-\frac {1}{x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \int \frac {\sqrt {1-\frac {1}{x}} x^3}{\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}d\sqrt {1-\frac {1}{x}}\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle -4 \int \frac {1}{\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (1-\frac {1}{x}\right )^{5/2} \left (1+\frac {1}{x}\right )^3 x}d\sqrt {1-\sqrt {1-\frac {1}{x}}}\) |
| \(\Big \downarrow \) 2003 |
| \(\displaystyle -4 \int \frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (\sqrt {1-\sqrt {1-\frac {1}{x}}}+1\right )}{\left (1-\frac {1}{x}\right )^{5/2} \left (1+\frac {1}{x}\right )^3}d\sqrt {1-\sqrt {1-\frac {1}{x}}}\) |
| \(\Big \downarrow \) 2353 |
| \(\displaystyle -4 \int \left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (-\sqrt {1-\sqrt {1-\frac {1}{x}}}-2\right )}{8 \left (-1-\frac {1}{x}\right )^3}+\frac {3 \sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{16 \sqrt {1-\frac {1}{x}}}-\frac {3 \sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (\sqrt {1-\sqrt {1-\frac {1}{x}}}+1\right )}{16 \left (-1-\frac {1}{x}\right )}+\frac {3 \sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{16 \left (1-\frac {1}{x}\right )}+\frac {3 \sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{16 \left (1-\frac {1}{x}\right )^{3/2}}+\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (3 \sqrt {1-\sqrt {1-\frac {1}{x}}}+4\right )}{16 \left (-1-\frac {1}{x}\right )^2}+\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{8 \left (1-\frac {1}{x}\right )^2}+\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{8 \left (1-\frac {1}{x}\right )^{5/2}}\right )d\sqrt {1-\sqrt {1-\frac {1}{x}}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -4 \left (-\frac {1}{64} \sqrt {\frac {1}{2} \left (1+29 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {\sqrt {2}-1}}\right )-\frac {3}{512} \sqrt {7+13 \sqrt {2}} \arctan \left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {\sqrt {2}-1}}\right )+\frac {3 \arctan \left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {\sqrt {2}-1}}\right )}{16 \sqrt {2 \left (\sqrt {2}-1\right )}}-\frac {59}{512} \text {arctanh}\left (\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )-\frac {1}{64} \sqrt {\frac {1}{2} \left (29 \sqrt {2}-1\right )} \text {arctanh}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {1+\sqrt {2}}}\right )-\frac {3}{512} \sqrt {13 \sqrt {2}-7} \text {arctanh}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {1+\sqrt {2}}}\right )+\frac {3 \text {arctanh}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {1+\sqrt {2}}}\right )}{16 \sqrt {2 \left (1+\sqrt {2}\right )}}-\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (4-3 \sqrt {1-\sqrt {1-\frac {1}{x}}}\right )}{256 \left (\frac {1}{x}+1\right )}-\frac {59 \sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{512 \sqrt {1-\frac {1}{x}}}-\frac {59 \sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{768 \left (1-\frac {1}{x}\right )}+\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (2 \sqrt {1-\sqrt {1-\frac {1}{x}}}+3\right )}{32 \left (\frac {1}{x}+1\right )}-\frac {7 \sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{192 \left (1-\frac {1}{x}\right )^{3/2}}-\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{32 \left (1-\frac {1}{x}\right )^2}+\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (\sqrt {1-\sqrt {1-\frac {1}{x}}}+1\right )}{32 \left (\frac {1}{x}+1\right )^2}\right )\) |
-4*(-1/32*Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]]/(1 - x^(-1))^2 - (7*Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]])/(192*(1 - x^(-1))^(3/2)) - (59*Sqrt[1 - Sqr t[1 - Sqrt[1 - x^(-1)]]])/(768*(1 - x^(-1))) - (59*Sqrt[1 - Sqrt[1 - Sqrt[ 1 - x^(-1)]]])/(512*Sqrt[1 - x^(-1)]) + (Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1) ]]]*(1 + Sqrt[1 - Sqrt[1 - x^(-1)]]))/(32*(1 + x^(-1))^2) - ((4 - 3*Sqrt[1 - Sqrt[1 - x^(-1)]])*Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]])/(256*(1 + x^(- 1))) + (Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]]*(3 + 2*Sqrt[1 - Sqrt[1 - x^(- 1)]]))/(32*(1 + x^(-1))) + (3*ArcTan[Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]]/ Sqrt[-1 + Sqrt[2]]])/(16*Sqrt[2*(-1 + Sqrt[2])]) - (3*Sqrt[7 + 13*Sqrt[2]] *ArcTan[Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]]/Sqrt[-1 + Sqrt[2]]])/512 - (S qrt[(1 + 29*Sqrt[2])/2]*ArcTan[Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]]/Sqrt[- 1 + Sqrt[2]]])/64 - (59*ArcTanh[Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]]])/512 + (3*ArcTanh[Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]]/Sqrt[1 + Sqrt[2]]])/(16 *Sqrt[2*(1 + Sqrt[2])]) - (3*Sqrt[-7 + 13*Sqrt[2]]*ArcTanh[Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]]/Sqrt[1 + Sqrt[2]]])/512 - (Sqrt[(-1 + 29*Sqrt[2])/2] *ArcTanh[Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]]/Sqrt[1 + Sqrt[2]]])/64)
3.31.73.3.1 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]]
\[\int \frac {x}{\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}d x\]
Time = 0.25 (sec) , antiderivative size = 358, normalized size of antiderivative = 0.71 \[ \int \frac {x}{\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}} \, dx=\frac {1}{384} \, {\left ({\left (16 \, x^{2} + {\left (208 \, x^{2} + 291 \, x\right )} \sqrt {\frac {x - 1}{x}} + 55 \, x\right )} \sqrt {-\sqrt {\frac {x - 1}{x}} + 1} + 2 \, {\left (96 \, x^{2} + 119 \, x\right )} \sqrt {\frac {x - 1}{x}} - 2 \, x\right )} \sqrt {-\sqrt {-\sqrt {\frac {x - 1}{x}} + 1} + 1} + \frac {1}{256} \, \sqrt {1021 \, \sqrt {2} - 1439} \log \left (\sqrt {1021 \, \sqrt {2} - 1439} {\left (30 \, \sqrt {2} + 41\right )} + 119 \, \sqrt {-\sqrt {-\sqrt {\frac {x - 1}{x}} + 1} + 1}\right ) - \frac {1}{256} \, \sqrt {1021 \, \sqrt {2} - 1439} \log \left (-\sqrt {1021 \, \sqrt {2} - 1439} {\left (30 \, \sqrt {2} + 41\right )} + 119 \, \sqrt {-\sqrt {-\sqrt {\frac {x - 1}{x}} + 1} + 1}\right ) - \frac {1}{256} \, \sqrt {-1021 \, \sqrt {2} - 1439} \log \left ({\left (30 \, \sqrt {2} - 41\right )} \sqrt {-1021 \, \sqrt {2} - 1439} + 119 \, \sqrt {-\sqrt {-\sqrt {\frac {x - 1}{x}} + 1} + 1}\right ) + \frac {1}{256} \, \sqrt {-1021 \, \sqrt {2} - 1439} \log \left (-{\left (30 \, \sqrt {2} - 41\right )} \sqrt {-1021 \, \sqrt {2} - 1439} + 119 \, \sqrt {-\sqrt {-\sqrt {\frac {x - 1}{x}} + 1} + 1}\right ) + \frac {59}{256} \, \log \left (\sqrt {-\sqrt {-\sqrt {\frac {x - 1}{x}} + 1} + 1} + 1\right ) - \frac {59}{256} \, \log \left (\sqrt {-\sqrt {-\sqrt {\frac {x - 1}{x}} + 1} + 1} - 1\right ) \]
1/384*((16*x^2 + (208*x^2 + 291*x)*sqrt((x - 1)/x) + 55*x)*sqrt(-sqrt((x - 1)/x) + 1) + 2*(96*x^2 + 119*x)*sqrt((x - 1)/x) - 2*x)*sqrt(-sqrt(-sqrt(( x - 1)/x) + 1) + 1) + 1/256*sqrt(1021*sqrt(2) - 1439)*log(sqrt(1021*sqrt(2 ) - 1439)*(30*sqrt(2) + 41) + 119*sqrt(-sqrt(-sqrt((x - 1)/x) + 1) + 1)) - 1/256*sqrt(1021*sqrt(2) - 1439)*log(-sqrt(1021*sqrt(2) - 1439)*(30*sqrt(2 ) + 41) + 119*sqrt(-sqrt(-sqrt((x - 1)/x) + 1) + 1)) - 1/256*sqrt(-1021*sq rt(2) - 1439)*log((30*sqrt(2) - 41)*sqrt(-1021*sqrt(2) - 1439) + 119*sqrt( -sqrt(-sqrt((x - 1)/x) + 1) + 1)) + 1/256*sqrt(-1021*sqrt(2) - 1439)*log(- (30*sqrt(2) - 41)*sqrt(-1021*sqrt(2) - 1439) + 119*sqrt(-sqrt(-sqrt((x - 1 )/x) + 1) + 1)) + 59/256*log(sqrt(-sqrt(-sqrt((x - 1)/x) + 1) + 1) + 1) - 59/256*log(sqrt(-sqrt(-sqrt((x - 1)/x) + 1) + 1) - 1)
\[ \int \frac {x}{\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}} \, dx=\int \frac {x}{\sqrt {1 - \sqrt {1 - \sqrt {1 - \frac {1}{x}}}}}\, dx \]
\[ \int \frac {x}{\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}} \, dx=\int { \frac {x}{\sqrt {-\sqrt {-\sqrt {-\frac {1}{x} + 1} + 1} + 1}} \,d x } \]
Timed out. \[ \int \frac {x}{\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {x}{\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}} \, dx=\int \frac {x}{\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}} \,d x \]