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

Optimal. Leaf size=172 \[ -4 \sqrt {1-\sqrt {1-\sqrt {\frac {x^2-1}{x^2}}}}+\sqrt {\sqrt {2}-1} \tan ^{-1}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {\frac {x^2-1}{x^2}}}}}{\sqrt {\sqrt {2}-1}}\right )+2 \tanh ^{-1}\left (\sqrt {1-\sqrt {1-\sqrt {\frac {x^2-1}{x^2}}}}\right )+\sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {\frac {x^2-1}{x^2}}}}}{\sqrt {1+\sqrt {2}}}\right ) \]

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Rubi [A]  time = 1.18, antiderivative size = 164, normalized size of antiderivative = 0.95, number of steps used = 12, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {6742, 2073, 207, 1166, 203} \begin {gather*} -4 \sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x^2}}}}+\sqrt {\sqrt {2}-1} \tan ^{-1}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x^2}}}}}{\sqrt {\sqrt {2}-1}}\right )+2 \tanh ^{-1}\left (\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x^2}}}}\right )+\sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x^2}}}}}{\sqrt {1+\sqrt {2}}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-4*Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-2)]]] + Sqrt[-1 + Sqrt[2]]*ArcTan[Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-2)]]]/Sqrt[
-1 + Sqrt[2]]] + 2*ArcTanh[Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-2)]]]] + Sqrt[1 + Sqrt[2]]*ArcTanh[Sqrt[1 - Sqrt[1 -
Sqrt[1 - x^(-2)]]]/Sqrt[1 + Sqrt[2]]]

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 2073

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P /. x -> Sqrt[x]]}, Int[ExpandIntegrand[(PP /. x ->
x^2)^p*Q^q, x], x] /;  !SumQ[NonfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x^2] && PolyQ[Q, x] && ILtQ[p,
0]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x^2}}}}}{x} \, dx &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt {1-\sqrt {1-\sqrt {1-x}}}}{x} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=\operatorname {Subst}\left (\int \frac {\sqrt {1-\sqrt {1-x}} x}{1-x^2} \, dx,x,\sqrt {1-\frac {1}{x^2}}\right )\\ &=-\left (2 \operatorname {Subst}\left (\int \frac {\sqrt {1-x} \left (-1+x^2\right )}{x \left (-2+x^2\right )} \, dx,x,\sqrt {1-\sqrt {1-\frac {1}{x^2}}}\right )\right )\\ &=-\left (4 \operatorname {Subst}\left (\int \frac {x^4 \left (-2+x^2\right )}{1+x^2-3 x^4+x^6} \, dx,x,\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x^2}}}}\right )\right )\\ &=-\left (4 \operatorname {Subst}\left (\int \left (1-\frac {1+x^2-x^4}{1+x^2-3 x^4+x^6}\right ) \, dx,x,\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x^2}}}}\right )\right )\\ &=-4 \sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x^2}}}}+4 \operatorname {Subst}\left (\int \frac {1+x^2-x^4}{1+x^2-3 x^4+x^6} \, dx,x,\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x^2}}}}\right )\\ &=-4 \sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x^2}}}}+4 \operatorname {Subst}\left (\int \left (-\frac {1}{2 \left (-1+x^2\right )}+\frac {-1-x^2}{2 \left (-1-2 x^2+x^4\right )}\right ) \, dx,x,\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x^2}}}}\right )\\ &=-4 \sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x^2}}}}-2 \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x^2}}}}\right )+2 \operatorname {Subst}\left (\int \frac {-1-x^2}{-1-2 x^2+x^4} \, dx,x,\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x^2}}}}\right )\\ &=-4 \sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x^2}}}}+2 \tanh ^{-1}\left (\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x^2}}}}\right )+\left (-1-\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^2}}}}\right )+\left (-1+\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^2}}}}\right )\\ &=-4 \sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x^2}}}}+\sqrt {-1+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x^2}}}}}{\sqrt {-1+\sqrt {2}}}\right )+2 \tanh ^{-1}\left (\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x^2}}}}\right )+\sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x^2}}}}}{\sqrt {1+\sqrt {2}}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.32, size = 164, normalized size = 0.95 \begin {gather*} -4 \sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x^2}}}}+\sqrt {\sqrt {2}-1} \tan ^{-1}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x^2}}}}}{\sqrt {\sqrt {2}-1}}\right )+2 \tanh ^{-1}\left (\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x^2}}}}\right )+\sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x^2}}}}}{\sqrt {1+\sqrt {2}}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-4*Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-2)]]] + Sqrt[-1 + Sqrt[2]]*ArcTan[Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-2)]]]/Sqrt[
-1 + Sqrt[2]]] + 2*ArcTanh[Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-2)]]]] + Sqrt[1 + Sqrt[2]]*ArcTanh[Sqrt[1 - Sqrt[1 -
Sqrt[1 - x^(-2)]]]/Sqrt[1 + Sqrt[2]]]

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IntegrateAlgebraic [A]  time = 1.21, size = 172, normalized size = 1.00 \begin {gather*} -4 \sqrt {1-\sqrt {1-\sqrt {\frac {-1+x^2}{x^2}}}}+\sqrt {-1+\sqrt {2}} \tan ^{-1}\left (\sqrt {1+\sqrt {2}} \sqrt {1-\sqrt {1-\sqrt {\frac {-1+x^2}{x^2}}}}\right )+2 \tanh ^{-1}\left (\sqrt {1-\sqrt {1-\sqrt {\frac {-1+x^2}{x^2}}}}\right )+\sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\sqrt {-1+\sqrt {2}} \sqrt {1-\sqrt {1-\sqrt {\frac {-1+x^2}{x^2}}}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-4*Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x^2)/x^2]]] + Sqrt[-1 + Sqrt[2]]*ArcTan[Sqrt[1 + Sqrt[2]]*Sqrt[1 - Sqrt[1 - Sq
rt[(-1 + x^2)/x^2]]]] + 2*ArcTanh[Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x^2)/x^2]]]] + Sqrt[1 + Sqrt[2]]*ArcTanh[Sqrt[-
1 + Sqrt[2]]*Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x^2)/x^2]]]]

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fricas [B]  time = 146.08, size = 1037, normalized size = 6.03

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(sqrt(2) - 1)*arctan(-1/45218*(2*sqrt(2)*(21008*x^4 + 608*x^2 + sqrt(2)*(15192*x^4 - 163*x^2) + (32368*x^
4 + 248*x^2 + sqrt(2)*(22856*x^4 + 231*x^2))*sqrt((x^2 - 1)/x^2))*sqrt(141401*sqrt(2) - 198689)*sqrt(sqrt(2) -
1)*sqrt(-sqrt((x^2 - 1)/x^2) + 1) + sqrt(2)*(68704*x^4 - 76436*x^2 + sqrt(2)*(49408*x^4 - 55516*x^2 - 479) +
4*(9512*x^4 + 17*x^2 + sqrt(2)*(6672*x^4 + 107*x^2))*sqrt((x^2 - 1)/x^2) - 710)*sqrt(141401*sqrt(2) - 198689)*
sqrt(sqrt(2) - 1) - 90436*((40*x^4 - 5*x^2 + 4*sqrt(2)*(6*x^4 + x^2) + (40*sqrt(2)*x^4 + 56*x^4 + x^2)*sqrt((x
^2 - 1)/x^2))*sqrt(sqrt(2) - 1)*sqrt(-sqrt((x^2 - 1)/x^2) + 1) + (32*x^4 - 18*x^2 + sqrt(2)*(8*x^4 + 13*x^2) +
(32*x^4 + 2*x^2 + sqrt(2)*(24*x^4 - x^2))*sqrt((x^2 - 1)/x^2))*sqrt(sqrt(2) - 1))*sqrt(-sqrt(-sqrt((x^2 - 1)/
x^2) + 1) + 1))/(64*x^4 - 112*x^2 - 1)) + 1/4*sqrt(sqrt(2) + 1)*log(4*(479*sqrt(2)*x^2 + 710*x^2 - (479*sqrt(2
)*x^2 + 710*x^2)*sqrt((x^2 - 1)/x^2))*sqrt(sqrt(2) + 1)*sqrt(-sqrt((x^2 - 1)/x^2) + 1) - 2*(1916*x^2 + sqrt(2)
*(1420*x^2 - 231) - 4*(355*sqrt(2)*x^2 + 479*x^2)*sqrt((x^2 - 1)/x^2) - 248)*sqrt(sqrt(2) + 1) - 4*(710*sqrt(2
)*x^2 + 958*x^2 - (479*sqrt(2)*x^2 + 710*x^2 - (479*sqrt(2)*x^2 + 710*x^2)*sqrt((x^2 - 1)/x^2))*sqrt(-sqrt((x^
2 - 1)/x^2) + 1) - 2*(355*sqrt(2)*x^2 + 479*x^2)*sqrt((x^2 - 1)/x^2))*sqrt(-sqrt(-sqrt((x^2 - 1)/x^2) + 1) + 1
)) - 1/4*sqrt(sqrt(2) + 1)*log(-4*(479*sqrt(2)*x^2 + 710*x^2 - (479*sqrt(2)*x^2 + 710*x^2)*sqrt((x^2 - 1)/x^2)
)*sqrt(sqrt(2) + 1)*sqrt(-sqrt((x^2 - 1)/x^2) + 1) + 2*(1916*x^2 + sqrt(2)*(1420*x^2 - 231) - 4*(355*sqrt(2)*x
^2 + 479*x^2)*sqrt((x^2 - 1)/x^2) - 248)*sqrt(sqrt(2) + 1) - 4*(710*sqrt(2)*x^2 + 958*x^2 - (479*sqrt(2)*x^2 +
710*x^2 - (479*sqrt(2)*x^2 + 710*x^2)*sqrt((x^2 - 1)/x^2))*sqrt(-sqrt((x^2 - 1)/x^2) + 1) - 2*(355*sqrt(2)*x^
2 + 479*x^2)*sqrt((x^2 - 1)/x^2))*sqrt(-sqrt(-sqrt((x^2 - 1)/x^2) + 1) + 1)) - 4*sqrt(-sqrt(-sqrt((x^2 - 1)/x^
2) + 1) + 1) + log(-2*(x^2*sqrt((x^2 - 1)/x^2) + x^2)*sqrt(-sqrt(-sqrt((x^2 - 1)/x^2) + 1) + 1)*sqrt(-sqrt((x^
2 - 1)/x^2) + 1) - 2*(x^2*sqrt((x^2 - 1)/x^2) + x^2)*sqrt(-sqrt((x^2 - 1)/x^2) + 1) + 1)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(x^8-4*x^6+4*x^4-4*x^2)]Error index.cc index_gcd Error: Bad Argument ValueError index.cc index_gcd Error: Bad
Argument ValueError index.cc index_gcd Error: Bad Argument ValueError index.cc index_gcd Error: Bad Argument
ValueDiscontinuities at zeroes of x^8-4*x^6+4*x^4-4*x^2 were not checkedEvaluation time: 2.24Done

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maple [F]  time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x^{2}}}}}}{x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {1-\sqrt {1-\sqrt {1-\frac {1}{x^2}}}}}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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