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

Optimal. Leaf size=501 \[ \frac {-291 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{15/2}+2275 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{13/2}-6611 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{11/2}+8403 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{9/2}-3301 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{7/2}-2139 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{5/2}+1787 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{3/2}+69 \sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{384 \left (\left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^2-2 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )-1\right )^2 \left (1-\sqrt {1-\frac {1}{x}}\right )^2}+\left (-\frac {19}{64} \sqrt {\frac {1}{2} \left (\frac {1}{2}+\frac {1}{\sqrt {2}}\right )}-\frac {11}{64} \sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}}\right ) \tan ^{-1}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {\sqrt {2}-1}}\right )+\frac {59}{128} \tanh ^{-1}\left (\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )+\left (\frac {19}{64} \sqrt {\frac {1}{2} \left (\frac {1}{\sqrt {2}}-\frac {1}{2}\right )}-\frac {11}{64} \sqrt {\frac {1}{\sqrt {2}}-\frac {1}{2}}\right ) \tanh ^{-1}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {1+\sqrt {2}}}\right ) \]

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Rubi [A]  time = 1.50, antiderivative size = 758, normalized size of antiderivative = 1.51, number of steps used = 16, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, 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 )}{8 \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 )}{8 \left (\sqrt {1-\frac {1}{x}}+1\right )^2}+\frac {59}{256 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )}-\frac {59}{256 \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 )}{64 \left (\sqrt {1-\frac {1}{x}}+1\right )}+\frac {23}{256 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^2}-\frac {23}{256 \left (\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}+1\right )^2}+\frac {5}{192 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^3}-\frac {5}{192 \left (\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}+1\right )^3}+\frac {1}{128 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^4}-\frac {1}{128 \left (\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}+1\right )^4}-\frac {1}{128} \sqrt {527+373 \sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {\sqrt {2}-1}}\right )-\frac {\left (1+\sqrt {2}\right )^{3/2} \tan ^{-1}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {\sqrt {2}-1}}\right )}{16 \sqrt {2}}+\frac {59}{128} \tanh ^{-1}\left (\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )+\frac {1}{128} \sqrt {373 \sqrt {2}-527} \tanh ^{-1}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {1+\sqrt {2}}}\right )+\frac {\left (\sqrt {2}-1\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {1+\sqrt {2}}}\right )}{16 \sqrt {2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

1/(128*(1 - Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]])^4) + 5/(192*(1 - Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]])^3) +
23/(256*(1 - Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]])^2) + 59/(256*(1 - Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]])) -
1/(128*(1 + Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]])^4) - 5/(192*(1 + Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]])^3) -
23/(256*(1 + Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]])^2) - 59/(256*(1 + Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]])) -
(Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]]*(1 + Sqrt[1 - Sqrt[1 - x^(-1)]]))/(8*(1 + Sqrt[1 - x^(-1)])^2) - (Sqrt[1
- Sqrt[1 - Sqrt[1 - x^(-1)]]]*(1 + Sqrt[1 - Sqrt[1 - x^(-1)]]))/(8*(1 + Sqrt[1 - x^(-1)])) - (Sqrt[1 - Sqrt[1
- Sqrt[1 - x^(-1)]]]*(12 + 11*Sqrt[1 - Sqrt[1 - x^(-1)]]))/(64*(1 + Sqrt[1 - x^(-1)])) - ((1 + Sqrt[2])^(3/2)
*ArcTan[Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]]/Sqrt[-1 + Sqrt[2]]])/(16*Sqrt[2]) - (Sqrt[527 + 373*Sqrt[2]]*ArcT
an[Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]]/Sqrt[-1 + Sqrt[2]]])/128 + (59*ArcTanh[Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-
1)]]]])/128 + ((-1 + Sqrt[2])^(3/2)*ArcTanh[Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]]/Sqrt[1 + Sqrt[2]]])/(16*Sqrt[
2]) + (Sqrt[-527 + 373*Sqrt[2]]*ArcTanh[Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]]/Sqrt[1 + Sqrt[2]]])/128

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}{\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}} \, dx &=-\left (2 \operatorname {Subst}\left (\int \frac {x}{\sqrt {1-\sqrt {1-x}} \left (-1+x^2\right )^3} \, dx,x,\sqrt {1-\frac {1}{x}}\right )\right )\\ &=4 \operatorname {Subst}\left (\int \frac {1-x^2}{\sqrt {1-x} x^5 \left (-2+x^2\right )^3} \, dx,x,\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )\\ &=4 \operatorname {Subst}\left (\int \frac {\sqrt {1-x} (1+x)}{x^5 \left (-2+x^2\right )^3} \, dx,x,\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )\\ &=-\left (8 \operatorname {Subst}\left (\int \frac {x^2 \left (-2+x^2\right )}{\left (-1+x^2\right )^5 \left (-1-2 x^2+x^4\right )^3} \, dx,x,\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )\right )\\ &=-\left (8 \operatorname {Subst}\left (\int \left (\frac {1}{256 (-1+x)^5}-\frac {5}{512 (-1+x)^4}+\frac {23}{1024 (-1+x)^3}-\frac {59}{2048 (-1+x)^2}-\frac {1}{256 (1+x)^5}-\frac {5}{512 (1+x)^4}-\frac {23}{1024 (1+x)^3}-\frac {59}{2048 (1+x)^2}+\frac {59}{1024 \left (-1+x^2\right )}+\frac {-1+x^2}{8 \left (-1-2 x^2+x^4\right )^3}+\frac {1-x^2}{16 \left (-1-2 x^2+x^4\right )^2}\right ) \, dx,x,\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )\right )\\ &=\frac {1}{128 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^4}+\frac {5}{192 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^3}+\frac {23}{256 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^2}+\frac {59}{256 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )}-\frac {1}{128 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^4}-\frac {5}{192 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^3}-\frac {23}{256 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^2}-\frac {59}{256 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )}-\frac {59}{128} \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, 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 )^2} \, dx,x,\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )-\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}{128 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^4}+\frac {5}{192 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^3}+\frac {23}{256 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^2}+\frac {59}{256 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )}-\frac {1}{128 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^4}-\frac {5}{192 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^3}-\frac {23}{256 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^2}-\frac {59}{256 \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 )}{8 \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 )}{8 \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 {59}{128} \tanh ^{-1}\left (\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )-\frac {1}{32} \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}{32} \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}{128 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^4}+\frac {5}{192 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^3}+\frac {23}{256 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^2}+\frac {59}{256 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )}-\frac {1}{128 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^4}-\frac {5}{192 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^3}-\frac {23}{256 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^2}-\frac {59}{256 \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 )}{8 \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 )}{8 \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 )}{64 \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 {59}{128} \tanh ^{-1}\left (\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )-\frac {1}{512} \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 )-\frac {1}{32} \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}{32} \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}{128 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^4}+\frac {5}{192 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^3}+\frac {23}{256 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^2}+\frac {59}{256 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )}-\frac {1}{128 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^4}-\frac {5}{192 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^3}-\frac {23}{256 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^2}-\frac {59}{256 \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 )}{8 \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 )}{8 \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 )}{64 \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 )}{16 \sqrt {2}}+\frac {59}{128} \tanh ^{-1}\left (\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\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 )}{16 \sqrt {2}}-\frac {1}{128} \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}{128} \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}{128 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^4}+\frac {5}{192 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^3}+\frac {23}{256 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^2}+\frac {59}{256 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )}-\frac {1}{128 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^4}-\frac {5}{192 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^3}-\frac {23}{256 \left (1+\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^2}-\frac {59}{256 \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 )}{8 \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 )}{8 \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 )}{64 \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 )}{16 \sqrt {2}}-\frac {1}{128} \sqrt {527+373 \sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {-1+\sqrt {2}}}\right )+\frac {59}{128} \tanh ^{-1}\left (\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\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 )}{16 \sqrt {2}}+\frac {1}{128} \sqrt {-527+373 \sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {1+\sqrt {2}}}\right )\\ \end {align*}

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Mathematica [A]  time = 1.40, size = 467, normalized size = 0.93 \begin {gather*} \frac {32 \sqrt {1-\sqrt {\frac {x-1}{x}}} x^2+32 \sqrt {1-\sqrt {\frac {x-1}{x}}} \sqrt {\frac {x-1}{x}} x^2+384 x^2+114 \sqrt {1-\sqrt {\frac {x-1}{x}}} x+106 \sqrt {1-\sqrt {\frac {x-1}{x}}} \sqrt {\frac {x-1}{x}} x+4 \sqrt {\frac {x-1}{x}} x+52 x-177 \sqrt {1-\sqrt {1-\sqrt {\frac {x-1}{x}}}} \log \left (1-\frac {1}{\sqrt {1-\sqrt {1-\sqrt {\frac {x-1}{x}}}}}\right )+177 \sqrt {1-\sqrt {1-\sqrt {\frac {x-1}{x}}}} \log \left (\frac {1}{\sqrt {1-\sqrt {1-\sqrt {\frac {x-1}{x}}}}}+1\right )+6 \sqrt {\sqrt {2}-1} \left (41+30 \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 )+6 \sqrt {1+\sqrt {2}} \left (30 \sqrt {2}-41\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 )-582}{768 \sqrt {1-\sqrt {1-\sqrt {\frac {x-1}{x}}}}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-582 + 52*x + 114*Sqrt[1 - Sqrt[(-1 + x)/x]]*x + 4*Sqrt[(-1 + x)/x]*x + 106*Sqrt[1 - Sqrt[(-1 + x)/x]]*Sqrt[(
-1 + x)/x]*x + 384*x^2 + 32*Sqrt[1 - Sqrt[(-1 + x)/x]]*x^2 + 32*Sqrt[1 - Sqrt[(-1 + x)/x]]*Sqrt[(-1 + x)/x]*x^
2 + 6*Sqrt[-1 + Sqrt[2]]*(41 + 30*Sqrt[2])*Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x)/x]]]*ArcTan[1/(Sqrt[1 + Sqrt[2]]*Sq
rt[1 - Sqrt[1 - Sqrt[(-1 + x)/x]]])] + 6*Sqrt[1 + Sqrt[2]]*(-41 + 30*Sqrt[2])*Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x)/
x]]]*ArcTanh[1/(Sqrt[-1 + Sqrt[2]]*Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x)/x]]])] - 177*Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x
)/x]]]*Log[1 - 1/Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x)/x]]]] + 177*Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x)/x]]]*Log[1 + 1/Sq
rt[1 - Sqrt[1 - Sqrt[(-1 + x)/x]]]])/(768*Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x)/x]]])

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IntegrateAlgebraic [F]  time = 3.69, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

Could not integrate

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fricas [A]  time = 0.55, size = 344, normalized size = 0.69 \begin {gather*} \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}{64} \, \sqrt {1021 \, \sqrt {2} + 1439} \arctan \left (-\frac {1}{119} \, \sqrt {1021 \, \sqrt {2} + 1439} {\left (11 \, \sqrt {2} - 19\right )} \sqrt {\sqrt {2} - \sqrt {-\sqrt {\frac {x - 1}{x}} + 1}} + \frac {1}{119} \, \sqrt {1021 \, \sqrt {2} + 1439} {\left (11 \, \sqrt {2} - 19\right )} \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 (-\sqrt {1021 \, \sqrt {2} - 1439} {\left (30 \, \sqrt {2} + 41\right )} + 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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)*sqr
t((x - 1)/x) - 2*x)*sqrt(-sqrt(-sqrt((x - 1)/x) + 1) + 1) + 1/64*sqrt(1021*sqrt(2) + 1439)*arctan(-1/119*sqrt(
1021*sqrt(2) + 1439)*(11*sqrt(2) - 19)*sqrt(sqrt(2) - sqrt(-sqrt((x - 1)/x) + 1)) + 1/119*sqrt(1021*sqrt(2) +
1439)*(11*sqrt(2) - 19)*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) - 14
39)*log(-sqrt(1021*sqrt(2) - 1439)*(30*sqrt(2) + 41) + 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)

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

[Out]

Timed out

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maple [F]  time = 0.12, size = 0, normalized size = 0.00 \[\int \frac {x}{\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\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/(1-(1-(1-1/x)^(1/2))^(1/2))^(1/2),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x}{\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/(1 - (1 - (1 - 1/x)^(1/2))^(1/2))^(1/2),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\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/(1-(1-(1-1/x)**(1/2))**(1/2))**(1/2),x)

[Out]

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

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