3.32.9 \(\int \frac {x^2}{\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}} \, dx\)

Optimal. Leaf size=617 \[ \frac {-17265 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{23/2}+204165 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{21/2}-1002573 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{19/2}+2613381 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{17/2}-3755230 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{15/2}+2531670 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{13/2}+215274 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{11/2}-1429914 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{9/2}+598095 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{7/2}+184677 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{5/2}-148125 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{3/2}-4395 \sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{30720 \left (\left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^2-2 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )-1\right )^3 \left (1-\sqrt {1-\frac {1}{x}}\right )^3}+\left (-\frac {7}{32} \sqrt {\frac {1}{2} \left (\frac {1}{2}+\frac {1}{\sqrt {2}}\right )}-\frac {137 \sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}}}{1024}\right ) \tan ^{-1}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {\sqrt {2}-1}}\right )+\frac {703 \tanh ^{-1}\left (\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )}{2048}+\left (\frac {7}{32} \sqrt {\frac {1}{2} \left (\frac {1}{\sqrt {2}}-\frac {1}{2}\right )}-\frac {137 \sqrt {\frac {1}{\sqrt {2}}-\frac {1}{2}}}{1024}\right ) \tanh ^{-1}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {1+\sqrt {2}}}\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 2.03, antiderivative size = 1229, normalized size of antiderivative = 1.99, number of steps used = 22, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {1586, 1692, 207, 1178, 1166, 203} \begin {gather*} -\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (\sqrt {1-\sqrt {1-\frac {1}{x}}}+1\right )}{16 \left (\sqrt {1-\frac {1}{x}}+1\right )}-\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (\sqrt {1-\sqrt {1-\frac {1}{x}}}+1\right )}{16 \left (\sqrt {1-\frac {1}{x}}+1\right )^2}-\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (\sqrt {1-\sqrt {1-\frac {1}{x}}}+1\right )}{24 \left (\sqrt {1-\frac {1}{x}}+1\right )^3}-\frac {\sqrt {\frac {1}{2} \left (12049+8521 \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {-1+\sqrt {2}}}\right )}{1024}-\frac {1}{256} \sqrt {527+373 \sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {-1+\sqrt {2}}}\right )-\frac {\left (1+\sqrt {2}\right )^{3/2} \tan ^{-1}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {-1+\sqrt {2}}}\right )}{32 \sqrt {2}}+\frac {703 \tanh ^{-1}\left (\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )}{2048}+\frac {\sqrt {\frac {1}{2} \left (-12049+8521 \sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {1+\sqrt {2}}}\right )}{1024}+\frac {1}{256} \sqrt {-527+373 \sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {1+\sqrt {2}}}\right )+\frac {\left (-1+\sqrt {2}\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {1+\sqrt {2}}}\right )}{32 \sqrt {2}}+\frac {703}{4096 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )}-\frac {703}{4096 \left (\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}+1\right )}-\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (11 \sqrt {1-\sqrt {1-\frac {1}{x}}}+12\right )}{128 \left (\sqrt {1-\frac {1}{x}}+1\right )}-\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (108 \sqrt {1-\sqrt {1-\frac {1}{x}}}+121\right )}{1536 \left (\sqrt {1-\frac {1}{x}}+1\right )}+\frac {289}{4096 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^2}-\frac {289}{4096 \left (\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}+1\right )^2}-\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (19 \sqrt {1-\sqrt {1-\frac {1}{x}}}+20\right )}{384 \left (\sqrt {1-\frac {1}{x}}+1\right )^2}+\frac {41}{1536 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^3}-\frac {41}{1536 \left (\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}+1\right )^3}+\frac {11}{1024 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^4}-\frac {11}{1024 \left (\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}+1\right )^4}+\frac {7}{2560 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^5}-\frac {7}{2560 \left (\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}+1\right )^5}+\frac {1}{1536 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^6}-\frac {1}{1536 \left (\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}+1\right )^6} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[x^2/Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]],x]

[Out]

1/(1536*(1 - Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]])^6) + 7/(2560*(1 - Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]])^5)
+ 11/(1024*(1 - Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]])^4) + 41/(1536*(1 - Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]])
^3) + 289/(4096*(1 - Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]])^2) + 703/(4096*(1 - Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-
1)]]])) - 1/(1536*(1 + Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]])^6) - 7/(2560*(1 + Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-
1)]]])^5) - 11/(1024*(1 + Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]])^4) - 41/(1536*(1 + Sqrt[1 - Sqrt[1 - Sqrt[1 -
x^(-1)]]])^3) - 289/(4096*(1 + Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]])^2) - 703/(4096*(1 + Sqrt[1 - Sqrt[1 - Sqr
t[1 - x^(-1)]]])) - (Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]]*(1 + Sqrt[1 - Sqrt[1 - x^(-1)]]))/(24*(1 + Sqrt[1 -
x^(-1)])^3) - (Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]]*(1 + Sqrt[1 - Sqrt[1 - x^(-1)]]))/(16*(1 + Sqrt[1 - x^(-1)
])^2) - (Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]]*(20 + 19*Sqrt[1 - Sqrt[1 - x^(-1)]]))/(384*(1 + Sqrt[1 - x^(-1)]
)^2) - (Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]]*(1 + Sqrt[1 - Sqrt[1 - x^(-1)]]))/(16*(1 + Sqrt[1 - x^(-1)])) - (
Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]]*(12 + 11*Sqrt[1 - Sqrt[1 - x^(-1)]]))/(128*(1 + Sqrt[1 - x^(-1)])) - (Sqr
t[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]]*(121 + 108*Sqrt[1 - Sqrt[1 - x^(-1)]]))/(1536*(1 + Sqrt[1 - x^(-1)])) - ((1
+ Sqrt[2])^(3/2)*ArcTan[Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]]/Sqrt[-1 + Sqrt[2]]])/(32*Sqrt[2]) - (Sqrt[527 + 3
73*Sqrt[2]]*ArcTan[Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]]/Sqrt[-1 + Sqrt[2]]])/256 - (Sqrt[(12049 + 8521*Sqrt[2]
)/2]*ArcTan[Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]]/Sqrt[-1 + Sqrt[2]]])/1024 + (703*ArcTanh[Sqrt[1 - Sqrt[1 - Sq
rt[1 - x^(-1)]]]])/2048 + ((-1 + Sqrt[2])^(3/2)*ArcTanh[Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]]/Sqrt[1 + Sqrt[2]]
])/(32*Sqrt[2]) + (Sqrt[-527 + 373*Sqrt[2]]*ArcTanh[Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]]/Sqrt[1 + Sqrt[2]]])/2
56 + (Sqrt[(-12049 + 8521*Sqrt[2])/2]*ArcTanh[Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]]/Sqrt[1 + Sqrt[2]]])/1024

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 1178

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(x*(a*b*e - d*(b^2 - 2*
a*c) - c*(b*d - 2*a*e)*x^2)*(a + b*x^2 + c*x^4)^(p + 1))/(2*a*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*a*(p + 1)
*(b^2 - 4*a*c)), Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a +
b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e
^2, 0] && LtQ[p, -1] && IntegerQ[2*p]

Rule 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rule 1692

Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandInteg
rand[Px*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, q}, x] && PolyQ[Px, x^2] && NeQ[b
^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {x^2}{\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}} \, dx &=2 \operatorname {Subst}\left (\int \frac {x}{\sqrt {1-\sqrt {1-x}} \left (-1+x^2\right )^4} \, dx,x,\sqrt {1-\frac {1}{x}}\right )\\ &=-\left (4 \operatorname {Subst}\left (\int \frac {1-x^2}{\sqrt {1-x} x^7 \left (-2+x^2\right )^4} \, dx,x,\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )\right )\\ &=-\left (4 \operatorname {Subst}\left (\int \frac {\sqrt {1-x} (1+x)}{x^7 \left (-2+x^2\right )^4} \, dx,x,\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )\right )\\ &=8 \operatorname {Subst}\left (\int \frac {x^2 \left (-2+x^2\right )}{\left (-1+x^2\right )^7 \left (-1-2 x^2+x^4\right )^4} \, dx,x,\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )\\ &=8 \operatorname {Subst}\left (\int \left (-\frac {1}{2048 (-1+x)^7}+\frac {7}{4096 (-1+x)^6}-\frac {11}{2048 (-1+x)^5}+\frac {41}{4096 (-1+x)^4}-\frac {289}{16384 (-1+x)^3}+\frac {703}{32768 (-1+x)^2}+\frac {1}{2048 (1+x)^7}+\frac {7}{4096 (1+x)^6}+\frac {11}{2048 (1+x)^5}+\frac {41}{4096 (1+x)^4}+\frac {289}{16384 (1+x)^3}+\frac {703}{32768 (1+x)^2}-\frac {703}{16384 \left (-1+x^2\right )}+\frac {-1+x^2}{16 \left (-1-2 x^2+x^4\right )^4}+\frac {1-x^2}{16 \left (-1-2 x^2+x^4\right )^3}+\frac {-1+x^2}{32 \left (-1-2 x^2+x^4\right )^2}\right ) \, dx,x,\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )\\ &=\frac {1}{1536 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^6}+\frac {7}{2560 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^5}+\frac {11}{1024 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^4}+\frac {41}{1536 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^3}+\frac {289}{4096 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^2}+\frac {703}{4096 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )}-\frac {1}{1536 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^6}-\frac {7}{2560 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^5}-\frac {11}{1024 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^4}-\frac {41}{1536 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^3}-\frac {289}{4096 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^2}-\frac {703}{4096 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )}+\frac {1}{4} \operatorname {Subst}\left (\int \frac {-1+x^2}{\left (-1-2 x^2+x^4\right )^2} \, dx,x,\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )-\frac {703 \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )}{2048}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {-1+x^2}{\left (-1-2 x^2+x^4\right )^4} \, dx,x,\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1-x^2}{\left (-1-2 x^2+x^4\right )^3} \, dx,x,\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )\\ &=\frac {1}{1536 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^6}+\frac {7}{2560 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^5}+\frac {11}{1024 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^4}+\frac {41}{1536 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^3}+\frac {289}{4096 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^2}+\frac {703}{4096 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )}-\frac {1}{1536 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^6}-\frac {7}{2560 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^5}-\frac {11}{1024 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^4}-\frac {41}{1536 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^3}-\frac {289}{4096 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^2}-\frac {703}{4096 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )}-\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (1+\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )}{24 \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}-\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (1+\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )}{16 \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}-\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (1+\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )}{16 \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 )}+\frac {703 \tanh ^{-1}\left (\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )}{2048}+\frac {1}{96} \operatorname {Subst}\left (\int \frac {40-36 x^2}{\left (-1-2 x^2+x^4\right )^3} \, dx,x,\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )+\frac {1}{64} \operatorname {Subst}\left (\int \frac {-24+20 x^2}{\left (-1-2 x^2+x^4\right )^2} \, dx,x,\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )+\frac {1}{64} \operatorname {Subst}\left (\int \frac {8-4 x^2}{-1-2 x^2+x^4} \, dx,x,\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )\\ &=\frac {1}{1536 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^6}+\frac {7}{2560 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^5}+\frac {11}{1024 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^4}+\frac {41}{1536 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^3}+\frac {289}{4096 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^2}+\frac {703}{4096 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )}-\frac {1}{1536 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^6}-\frac {7}{2560 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^5}-\frac {11}{1024 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^4}-\frac {41}{1536 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^3}-\frac {289}{4096 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^2}-\frac {703}{4096 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )}-\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (1+\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )}{24 \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}-\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (1+\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )}{16 \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}-\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (1+\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )}{16 \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 )}-\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (12+11 \sqrt {1-\sqrt {-\frac {1-x}{x}}}\right )}{128 \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 )}-\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (20+19 \sqrt {1-\sqrt {-\frac {1-x}{x}}}\right )}{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}+\frac {703 \tanh ^{-1}\left (\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )}{2048}+\frac {\operatorname {Subst}\left (\int \frac {-968+760 x^2}{\left (-1-2 x^2+x^4\right )^2} \, dx,x,\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )}{3072}+\frac {\operatorname {Subst}\left (\int \frac {200-88 x^2}{-1-2 x^2+x^4} \, dx,x,\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )}{1024}+\frac {1}{64} \left (-2+\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-\sqrt {2}+x^2} \, dx,x,\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )-\frac {1}{64} \left (2+\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{-1+\sqrt {2}+x^2} \, dx,x,\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )\\ &=\frac {1}{1536 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^6}+\frac {7}{2560 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^5}+\frac {11}{1024 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^4}+\frac {41}{1536 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^3}+\frac {289}{4096 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^2}+\frac {703}{4096 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )}-\frac {1}{1536 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^6}-\frac {7}{2560 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^5}-\frac {11}{1024 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^4}-\frac {41}{1536 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^3}-\frac {289}{4096 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^2}-\frac {703}{4096 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )}-\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (1+\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )}{24 \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}-\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (1+\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )}{16 \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}-\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (1+\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )}{16 \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 )}-\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (12+11 \sqrt {1-\sqrt {-\frac {1-x}{x}}}\right )}{128 \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 )}-\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (20+19 \sqrt {1-\sqrt {-\frac {1-x}{x}}}\right )}{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}-\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (121+108 \sqrt {1-\sqrt {-\frac {1-x}{x}}}\right )}{1536 \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 )}-\frac {\left (1+\sqrt {2}\right )^{3/2} \tan ^{-1}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {-1+\sqrt {2}}}\right )}{32 \sqrt {2}}+\frac {703 \tanh ^{-1}\left (\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )}{2048}+\frac {\left (-1+\sqrt {2}\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {1+\sqrt {2}}}\right )}{32 \sqrt {2}}+\frac {\operatorname {Subst}\left (\int \frac {8160-3456 x^2}{-1-2 x^2+x^4} \, dx,x,\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )}{49152}+\frac {1}{256} \left (-11-7 \sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{-1+\sqrt {2}+x^2} \, dx,x,\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )+\frac {1}{256} \left (-11+7 \sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-\sqrt {2}+x^2} \, dx,x,\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )\\ &=\frac {1}{1536 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^6}+\frac {7}{2560 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^5}+\frac {11}{1024 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^4}+\frac {41}{1536 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^3}+\frac {289}{4096 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^2}+\frac {703}{4096 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )}-\frac {1}{1536 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^6}-\frac {7}{2560 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^5}-\frac {11}{1024 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^4}-\frac {41}{1536 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^3}-\frac {289}{4096 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^2}-\frac {703}{4096 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )}-\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (1+\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )}{24 \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}-\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (1+\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )}{16 \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}-\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (1+\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )}{16 \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 )}-\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (12+11 \sqrt {1-\sqrt {-\frac {1-x}{x}}}\right )}{128 \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 )}-\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (20+19 \sqrt {1-\sqrt {-\frac {1-x}{x}}}\right )}{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}-\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (121+108 \sqrt {1-\sqrt {-\frac {1-x}{x}}}\right )}{1536 \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 )}-\frac {\left (1+\sqrt {2}\right )^{3/2} \tan ^{-1}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {-1+\sqrt {2}}}\right )}{32 \sqrt {2}}-\frac {1}{256} \sqrt {527+373 \sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {-1+\sqrt {2}}}\right )+\frac {703 \tanh ^{-1}\left (\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )}{2048}+\frac {\left (-1+\sqrt {2}\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {1+\sqrt {2}}}\right )}{32 \sqrt {2}}+\frac {1}{256} \sqrt {-527+373 \sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {1+\sqrt {2}}}\right )+\frac {\left (-72+49 \sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-\sqrt {2}+x^2} \, dx,x,\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )}{2048}-\frac {\left (72+49 \sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{-1+\sqrt {2}+x^2} \, dx,x,\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )}{2048}\\ &=\frac {1}{1536 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^6}+\frac {7}{2560 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^5}+\frac {11}{1024 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^4}+\frac {41}{1536 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^3}+\frac {289}{4096 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^2}+\frac {703}{4096 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )}-\frac {1}{1536 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^6}-\frac {7}{2560 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^5}-\frac {11}{1024 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^4}-\frac {41}{1536 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^3}-\frac {289}{4096 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^2}-\frac {703}{4096 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )}-\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (1+\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )}{24 \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}-\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (1+\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )}{16 \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}-\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (1+\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )}{16 \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 )}-\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (12+11 \sqrt {1-\sqrt {-\frac {1-x}{x}}}\right )}{128 \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 )}-\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (20+19 \sqrt {1-\sqrt {-\frac {1-x}{x}}}\right )}{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}-\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (121+108 \sqrt {1-\sqrt {-\frac {1-x}{x}}}\right )}{1536 \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 )}-\frac {\left (1+\sqrt {2}\right )^{3/2} \tan ^{-1}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {-1+\sqrt {2}}}\right )}{32 \sqrt {2}}-\frac {1}{256} \sqrt {527+373 \sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {-1+\sqrt {2}}}\right )-\frac {\sqrt {24098+17042 \sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {-1+\sqrt {2}}}\right )}{2048}+\frac {703 \tanh ^{-1}\left (\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )}{2048}+\frac {\left (-1+\sqrt {2}\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {1+\sqrt {2}}}\right )}{32 \sqrt {2}}+\frac {1}{256} \sqrt {-527+373 \sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {1+\sqrt {2}}}\right )+\frac {\sqrt {-24098+17042 \sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {1+\sqrt {2}}}\right )}{2048}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 1.98, size = 547, normalized size = 0.89 \begin {gather*} \frac {1024 \sqrt {1-\sqrt {\frac {x-1}{x}}} x^3+1024 \sqrt {1-\sqrt {\frac {x-1}{x}}} \sqrt {\frac {x-1}{x}} x^3+20480 x^3+1888 \sqrt {1-\sqrt {\frac {x-1}{x}}} x^2+1760 \sqrt {1-\sqrt {\frac {x-1}{x}}} \sqrt {\frac {x-1}{x}} x^2+64 \sqrt {\frac {x-1}{x}} x^2+1344 x^2+6702 \sqrt {1-\sqrt {\frac {x-1}{x}}} x+6030 \sqrt {1-\sqrt {\frac {x-1}{x}}} \sqrt {\frac {x-1}{x}} x+324 \sqrt {\frac {x-1}{x}} x+3092 x-10545 \sqrt {1-\sqrt {1-\sqrt {\frac {x-1}{x}}}} \log \left (1-\frac {1}{\sqrt {1-\sqrt {1-\sqrt {\frac {x-1}{x}}}}}\right )+10545 \sqrt {1-\sqrt {1-\sqrt {\frac {x-1}{x}}}} \log \left (\frac {1}{\sqrt {1-\sqrt {1-\sqrt {\frac {x-1}{x}}}}}+1\right )+30 \sqrt {\sqrt {2}-1} \left (498+361 \sqrt {2}\right ) \sqrt {1-\sqrt {1-\sqrt {\frac {x-1}{x}}}} \tan ^{-1}\left (\frac {1}{\sqrt {1+\sqrt {2}} \sqrt {1-\sqrt {1-\sqrt {\frac {x-1}{x}}}}}\right )+30 \sqrt {1+\sqrt {2}} \left (361 \sqrt {2}-498\right ) \sqrt {1-\sqrt {1-\sqrt {\frac {x-1}{x}}}} \tanh ^{-1}\left (\frac {1}{\sqrt {\sqrt {2}-1} \sqrt {1-\sqrt {1-\sqrt {\frac {x-1}{x}}}}}\right )-34530}{61440 \sqrt {1-\sqrt {1-\sqrt {\frac {x-1}{x}}}}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[x^2/Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]],x]

[Out]

(-34530 + 3092*x + 6702*Sqrt[1 - Sqrt[(-1 + x)/x]]*x + 324*Sqrt[(-1 + x)/x]*x + 6030*Sqrt[1 - Sqrt[(-1 + x)/x]
]*Sqrt[(-1 + x)/x]*x + 1344*x^2 + 1888*Sqrt[1 - Sqrt[(-1 + x)/x]]*x^2 + 64*Sqrt[(-1 + x)/x]*x^2 + 1760*Sqrt[1
- Sqrt[(-1 + x)/x]]*Sqrt[(-1 + x)/x]*x^2 + 20480*x^3 + 1024*Sqrt[1 - Sqrt[(-1 + x)/x]]*x^3 + 1024*Sqrt[1 - Sqr
t[(-1 + x)/x]]*Sqrt[(-1 + x)/x]*x^3 + 30*Sqrt[-1 + Sqrt[2]]*(498 + 361*Sqrt[2])*Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x
)/x]]]*ArcTan[1/(Sqrt[1 + Sqrt[2]]*Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x)/x]]])] + 30*Sqrt[1 + Sqrt[2]]*(-498 + 361*S
qrt[2])*Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x)/x]]]*ArcTanh[1/(Sqrt[-1 + Sqrt[2]]*Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x)/x]]
])] - 10545*Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x)/x]]]*Log[1 - 1/Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x)/x]]]] + 10545*Sqrt[
1 - Sqrt[1 - Sqrt[(-1 + x)/x]]]*Log[1 + 1/Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x)/x]]]])/(61440*Sqrt[1 - Sqrt[1 - Sqrt
[(-1 + x)/x]]])

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 3.70, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2}{\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^2/Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]],x]

[Out]

Could not integrate

________________________________________________________________________________________

fricas [A]  time = 0.52, size = 374, normalized size = 0.61 \begin {gather*} -\frac {1}{1024} \, \sqrt {2} \sqrt {74545 \, \sqrt {2} + 105233} \arctan \left (-\frac {1}{6319} \, \sqrt {74545 \, \sqrt {2} + 105233} {\left (112 \, \sqrt {2} - 137\right )} \sqrt {\sqrt {2} - \sqrt {-\sqrt {\frac {x - 1}{x}} + 1}} + \frac {1}{6319} \, \sqrt {74545 \, \sqrt {2} + 105233} {\left (112 \, \sqrt {2} - 137\right )} \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 (-\sqrt {74545 \, \sqrt {2} - 105233} {\left (249 \, \sqrt {2} + 361\right )} + 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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1024*sqrt(2)*sqrt(74545*sqrt(2) + 105233)*arctan(-1/6319*sqrt(74545*sqrt(2) + 105233)*(112*sqrt(2) - 137)*s
qrt(sqrt(2) - sqrt(-sqrt((x - 1)/x) + 1)) + 1/6319*sqrt(74545*sqrt(2) + 105233)*(112*sqrt(2) - 137)*sqrt(-sqrt
(-sqrt((x - 1)/x) + 1) + 1)) + 1/4096*sqrt(2)*sqrt(74545*sqrt(2) - 105233)*log(sqrt(74545*sqrt(2) - 105233)*(2
49*sqrt(2) + 361) + 6319*sqrt(-sqrt(-sqrt((x - 1)/x) + 1) + 1)) - 1/4096*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)) - 1/30720*
(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 + 7125*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
)

________________________________________________________________________________________

giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

maple [F]  time = 0.11, size = 0, normalized size = 0.00 \[\int \frac {x^{2}}{\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\sqrt {-\sqrt {-\sqrt {-\frac {1}{x} + 1} + 1} + 1}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2/sqrt(-sqrt(-sqrt(-1/x + 1) + 1) + 1), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^2}{\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\sqrt {1 - \sqrt {1 - \sqrt {1 - \frac {1}{x}}}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2/sqrt(1 - sqrt(1 - sqrt(1 - 1/x))), x)

________________________________________________________________________________________