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

Optimal. Leaf size=160 \[ -4 \sqrt {\sqrt {1-\sqrt {\frac {1}{x^2}+1}}+1}+\sqrt {\sqrt {2}-1} \tan ^{-1}\left (\frac {\sqrt {\sqrt {1-\sqrt {\frac {x^2+1}{x^2}}}+1}}{\sqrt {\sqrt {2}-1}}\right )+2 \tanh ^{-1}\left (\sqrt {\sqrt {1-\sqrt {\frac {x^2+1}{x^2}}}+1}\right )+\sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {1-\sqrt {\frac {x^2+1}{x^2}}}+1}}{\sqrt {1+\sqrt {2}}}\right ) \]

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Rubi [A]  time = 1.03, antiderivative size = 148, normalized size of antiderivative = 0.92, number of steps used = 12, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6742, 2073, 207, 1166, 203} \begin {gather*} -4 \sqrt {\sqrt {1-\sqrt {\frac {1}{x^2}+1}}+1}+\sqrt {\sqrt {2}-1} \tan ^{-1}\left (\frac {\sqrt {\sqrt {1-\sqrt {\frac {1}{x^2}+1}}+1}}{\sqrt {\sqrt {2}-1}}\right )+2 \tanh ^{-1}\left (\sqrt {\sqrt {1-\sqrt {\frac {1}{x^2}+1}}+1}\right )+\sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {1-\sqrt {\frac {1}{x^2}+1}}+1}}{\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 = 0.26, size = 148, normalized size = 0.92 \begin {gather*} -4 \sqrt {\sqrt {1-\sqrt {\frac {1}{x^2}+1}}+1}+\sqrt {\sqrt {2}-1} \tan ^{-1}\left (\frac {\sqrt {\sqrt {1-\sqrt {\frac {1}{x^2}+1}}+1}}{\sqrt {\sqrt {2}-1}}\right )+2 \tanh ^{-1}\left (\sqrt {\sqrt {1-\sqrt {\frac {1}{x^2}+1}}+1}\right )+\sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {1-\sqrt {\frac {1}{x^2}+1}}+1}}{\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 = 0.89, size = 160, normalized size = 1.00 \begin {gather*} -4 \sqrt {1+\sqrt {1-\sqrt {1+\frac {1}{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)]]] + Sqrt[-1 + Sqrt[2]]*ArcTan[Sqrt[1 + Sqrt[2]]*Sqrt[1 + Sqrt[1 - Sqrt[(
1 + x^2)/x^2]]]] + 2*ArcTanh[Sqrt[1 + Sqrt[1 - Sqrt[(1 + x^2)/x^2]]]] + Sqrt[1 + Sqrt[2]]*ArcTanh[Sqrt[-1 + Sq
rt[2]]*Sqrt[1 + Sqrt[1 - Sqrt[(1 + x^2)/x^2]]]]

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fricas [B]  time = 127.73, size = 1028, normalized size = 6.42

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(1/2098*(2*sqrt(2)*(4432*x^4 - 132*x^2 + sqrt(2)*(3208*x^4 + 37*x^2) + (6832*x^4 - 52*
x^2 + sqrt(2)*(4824*x^4 - 49*x^2))*sqrt((x^2 + 1)/x^2))*sqrt(6301*sqrt(2) - 8849)*sqrt(sqrt(2) - 1)*sqrt(-sqrt
((x^2 + 1)/x^2) + 1) - sqrt(2)*(14496*x^4 + 16124*x^2 + sqrt(2)*(10432*x^4 + 11724*x^2 - 101) + 4*(2008*x^4 -
3*x^2 + sqrt(2)*(1408*x^4 - 23*x^2))*sqrt((x^2 + 1)/x^2) - 150)*sqrt(6301*sqrt(2) - 8849)*sqrt(sqrt(2) - 1) +
4196*((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(sq
rt(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 + sq
rt(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*(101*sqrt(2)*x^2 + 150*x^2 - (101*sqrt(2)*x^2 + 150*x^2)*sqrt(
(x^2 + 1)/x^2))*sqrt(sqrt(2) + 1)*sqrt(-sqrt((x^2 + 1)/x^2) + 1) + 2*(404*x^2 + sqrt(2)*(300*x^2 + 49) - 4*(75
*sqrt(2)*x^2 + 101*x^2)*sqrt((x^2 + 1)/x^2) + 52)*sqrt(sqrt(2) + 1) - 4*(150*sqrt(2)*x^2 + 202*x^2 + (101*sqrt
(2)*x^2 + 150*x^2 - (101*sqrt(2)*x^2 + 150*x^2)*sqrt((x^2 + 1)/x^2))*sqrt(-sqrt((x^2 + 1)/x^2) + 1) - 2*(75*sq
rt(2)*x^2 + 101*x^2)*sqrt((x^2 + 1)/x^2))*sqrt(sqrt(-sqrt((x^2 + 1)/x^2) + 1) + 1)) + 1/4*sqrt(sqrt(2) + 1)*lo
g(-4*(101*sqrt(2)*x^2 + 150*x^2 - (101*sqrt(2)*x^2 + 150*x^2)*sqrt((x^2 + 1)/x^2))*sqrt(sqrt(2) + 1)*sqrt(-sqr
t((x^2 + 1)/x^2) + 1) - 2*(404*x^2 + sqrt(2)*(300*x^2 + 49) - 4*(75*sqrt(2)*x^2 + 101*x^2)*sqrt((x^2 + 1)/x^2)
+ 52)*sqrt(sqrt(2) + 1) - 4*(150*sqrt(2)*x^2 + 202*x^2 + (101*sqrt(2)*x^2 + 150*x^2 - (101*sqrt(2)*x^2 + 150*
x^2)*sqrt((x^2 + 1)/x^2))*sqrt(-sqrt((x^2 + 1)/x^2) + 1) - 2*(75*sqrt(2)*x^2 + 101*x^2)*sqrt((x^2 + 1)/x^2))*s
qrt(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:Erro
r 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 ValueEvaluation time: 1.7Done

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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 {\sqrt {1-\sqrt {\frac {1}{x^2}+1}}+1}}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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