\(\int \frac {x}{\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}} \, dx\) [3073]

Optimal result

Integrand size = 29, antiderivative size = 501 \[ \int \frac {x}{\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}} \, dx=\frac {69 \sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}+1787 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{3/2}-2139 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{5/2}-3301 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{7/2}+8403 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{9/2}-6611 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{11/2}+2275 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{13/2}-291 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^{15/2}}{384 \left (-1-2 \left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )+\left (1-\sqrt {1-\sqrt {1-\frac {1}{x}}}\right )^2\right )^2 \left (1-\sqrt {1-\frac {1}{x}}\right )^2}+\left (-\frac {19}{64} \sqrt {\frac {1}{2} \left (\frac {1}{2}+\frac {1}{\sqrt {2}}\right )}-\frac {11}{64} \sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}}\right ) \arctan \left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {-1+\sqrt {2}}}\right )+\frac {59}{128} \text {arctanh}\left (\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )+\left (\frac {19}{64} \sqrt {\frac {1}{2} \left (-\frac {1}{2}+\frac {1}{\sqrt {2}}\right )}-\frac {11}{64} \sqrt {-\frac {1}{2}+\frac {1}{\sqrt {2}}}\right ) \text {arctanh}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {1+\sqrt {2}}}\right ) \]

[Out]

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))

Rubi [A] (verified)

Time = 0.96 (sec) , antiderivative size = 758, normalized size of antiderivative = 1.51, number of steps used = 16, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {1600, 1706, 213, 1192, 1180, 209} \[ \int \frac {x}{\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}} \, dx=-\frac {1}{128} \sqrt {527+373 \sqrt {2}} \arctan \left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {\sqrt {2}-1}}\right )-\frac {\left (1+\sqrt {2}\right )^{3/2} \arctan \left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {\sqrt {2}-1}}\right )}{16 \sqrt {2}}+\frac {59}{128} \text {arctanh}\left (\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )+\frac {1}{128} \sqrt {373 \sqrt {2}-527} \text {arctanh}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {1+\sqrt {2}}}\right )+\frac {\left (\sqrt {2}-1\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {1+\sqrt {2}}}\right )}{16 \sqrt {2}}-\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (\sqrt {1-\sqrt {1-\frac {1}{x}}}+1\right )}{8 \left (\sqrt {1-\frac {1}{x}}+1\right )}-\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (\sqrt {1-\sqrt {1-\frac {1}{x}}}+1\right )}{8 \left (\sqrt {1-\frac {1}{x}}+1\right )^2}+\frac {59}{256 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )}-\frac {59}{256 \left (\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}+1\right )}-\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}} \left (11 \sqrt {1-\sqrt {1-\frac {1}{x}}}+12\right )}{64 \left (\sqrt {1-\frac {1}{x}}+1\right )}+\frac {23}{256 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^2}-\frac {23}{256 \left (\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}+1\right )^2}+\frac {5}{192 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^3}-\frac {5}{192 \left (\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}+1\right )^3}+\frac {1}{128 \left (1-\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )^4}-\frac {1}{128 \left (\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}+1\right )^4} \]

[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 209

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

Rule 213

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

Rule 1180

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 1192

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 1600

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 1706

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*} \text {integral}& = -\left (2 \text {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 \text {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 \text {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 \text {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 \text {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} \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )-\frac {1}{2} \text {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 )-\text {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} \text {arctanh}\left (\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )-\frac {1}{32} \text {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} \text {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} \text {arctanh}\left (\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )-\frac {1}{512} \text {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 ) \text {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 ) \text {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} \arctan \left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {-1+\sqrt {2}}}\right )}{16 \sqrt {2}}+\frac {59}{128} \text {arctanh}\left (\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}\right )+\frac {\left (-1+\sqrt {2}\right )^{3/2} \text {arctanh}\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 ) \text {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 ) \text {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} \arctan \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}} \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 )+\frac {\left (-1+\sqrt {2}\right )^{3/2} \text {arctanh}\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}} \text {arctanh}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}}{\sqrt {1+\sqrt {2}}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 1.44 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.57 \[ \int \frac {x}{\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}} \, dx=\frac {1}{384} \left (\sqrt {1-\sqrt {1-\sqrt {\frac {-1+x}{x}}}} \sqrt {1-\sqrt {\frac {-1+x}{x}}} x \left (55+291 \sqrt {\frac {-1+x}{x}}+16 \left (1+13 \sqrt {\frac {-1+x}{x}}\right ) x\right )+2 \sqrt {1-\sqrt {1-\sqrt {\frac {-1+x}{x}}}} x \left (-1+119 \sqrt {\frac {-1+x}{x}}+96 \sqrt {\frac {-1+x}{x}} x\right )-3 \sqrt {1439+1021 \sqrt {2}} \arctan \left (\sqrt {1+\sqrt {2}} \sqrt {1-\sqrt {1-\sqrt {\frac {-1+x}{x}}}}\right )+177 \text {arctanh}\left (\sqrt {1-\sqrt {1-\sqrt {\frac {-1+x}{x}}}}\right )+\frac {357 \text {arctanh}\left (\sqrt {-1+\sqrt {2}} \sqrt {1-\sqrt {1-\sqrt {\frac {-1+x}{x}}}}\right )}{\sqrt {1439+1021 \sqrt {2}}}\right ) \]

[In]

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

[Out]

(Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x)/x]]]*Sqrt[1 - Sqrt[(-1 + x)/x]]*x*(55 + 291*Sqrt[(-1 + x)/x] + 16*(1 + 13*Sqr
t[(-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]]]])
/Sqrt[1439 + 1021*Sqrt[2]])/384

Maple [F]

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

[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)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 358, normalized size of antiderivative = 0.71 \[ \int \frac {x}{\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x}}}}} \, dx=\frac {1}{384} \, {\left ({\left (16 \, x^{2} + {\left (208 \, x^{2} + 291 \, x\right )} \sqrt {\frac {x - 1}{x}} + 55 \, x\right )} \sqrt {-\sqrt {\frac {x - 1}{x}} + 1} + 2 \, {\left (96 \, x^{2} + 119 \, x\right )} \sqrt {\frac {x - 1}{x}} - 2 \, x\right )} \sqrt {-\sqrt {-\sqrt {\frac {x - 1}{x}} + 1} + 1} + \frac {1}{256} \, \sqrt {1021 \, \sqrt {2} - 1439} \log \left (\sqrt {1021 \, \sqrt {2} - 1439} {\left (30 \, \sqrt {2} + 41\right )} + 119 \, \sqrt {-\sqrt {-\sqrt {\frac {x - 1}{x}} + 1} + 1}\right ) - \frac {1}{256} \, \sqrt {1021 \, \sqrt {2} - 1439} \log \left (-\sqrt {1021 \, \sqrt {2} - 1439} {\left (30 \, \sqrt {2} + 41\right )} + 119 \, \sqrt {-\sqrt {-\sqrt {\frac {x - 1}{x}} + 1} + 1}\right ) - \frac {1}{256} \, \sqrt {-1021 \, \sqrt {2} - 1439} \log \left ({\left (30 \, \sqrt {2} - 41\right )} \sqrt {-1021 \, \sqrt {2} - 1439} + 119 \, \sqrt {-\sqrt {-\sqrt {\frac {x - 1}{x}} + 1} + 1}\right ) + \frac {1}{256} \, \sqrt {-1021 \, \sqrt {2} - 1439} \log \left (-{\left (30 \, \sqrt {2} - 41\right )} \sqrt {-1021 \, \sqrt {2} - 1439} + 119 \, \sqrt {-\sqrt {-\sqrt {\frac {x - 1}{x}} + 1} + 1}\right ) + \frac {59}{256} \, \log \left (\sqrt {-\sqrt {-\sqrt {\frac {x - 1}{x}} + 1} + 1} + 1\right ) - \frac {59}{256} \, \log \left (\sqrt {-\sqrt {-\sqrt {\frac {x - 1}{x}} + 1} + 1} - 1\right ) \]

[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) + 1/256*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)*l
og(-sqrt(1021*sqrt(2) - 1439)*(30*sqrt(2) + 41) + 119*sqrt(-sqrt(-sqrt((x - 1)/x) + 1) + 1)) - 1/256*sqrt(-102
1*sqrt(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)

Sympy [F]

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

[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)

Maxima [F]

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

[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)

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[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)