3.2994 \(\int \frac{1}{\sqrt{1+\sqrt{\frac{1}{x}}}} \, dx\)

Optimal. Leaf size=58 \[ \sqrt{\sqrt{\frac{1}{x}}+1} x-\frac{3 \sqrt{\sqrt{\frac{1}{x}}+1}}{2 \sqrt{\frac{1}{x}}}+\frac{3}{2} \tanh ^{-1}\left (\sqrt{\sqrt{\frac{1}{x}}+1}\right ) \]

[Out]

(-3*Sqrt[1 + Sqrt[x^(-1)]])/(2*Sqrt[x^(-1)]) + Sqrt[1 + Sqrt[x^(-1)]]*x + (3*ArcTanh[Sqrt[1 + Sqrt[x^(-1)]]])/
2

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Rubi [A]  time = 0.0158496, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {255, 190, 51, 63, 207} \[ \sqrt{\sqrt{\frac{1}{x}}+1} x-\frac{3 \sqrt{\sqrt{\frac{1}{x}}+1}}{2 \sqrt{\frac{1}{x}}}+\frac{3}{2} \tanh ^{-1}\left (\sqrt{\sqrt{\frac{1}{x}}+1}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[1 + Sqrt[x^(-1)]],x]

[Out]

(-3*Sqrt[1 + Sqrt[x^(-1)]])/(2*Sqrt[x^(-1)]) + Sqrt[1 + Sqrt[x^(-1)]]*x + (3*ArcTanh[Sqrt[1 + Sqrt[x^(-1)]]])/
2

Rule 255

Int[((a_) + (b_.)*((c_.)*(x_)^(q_.))^(n_))^(p_.), x_Symbol] :> With[{k = Denominator[n]}, Subst[Int[(a + b*c^n
*x^(n*q))^p, x], x^(1/k), (c*x^q)^(1/k)/(c^(1/k)*(x^(1/k))^(q - 1))]] /; FreeQ[{a, b, c, p, q}, x] && Fraction
Q[n]

Rule 190

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(1/n - 1)*(a + b*x)^p, x], x, x^n], x] /
; FreeQ[{a, b, p}, x] && FractionQ[n] && IntegerQ[1/n]

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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])

Rubi steps

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

Mathematica [A]  time = 0.130816, size = 70, normalized size = 1.21 \[ \frac{1}{4} \left (2 \left (2-3 \sqrt{\frac{1}{x}}\right ) \sqrt{\sqrt{\frac{1}{x}}+1} x-3 \log \left (1-\frac{1}{\sqrt{\sqrt{\frac{1}{x}}+1}}\right )+3 \log \left (\frac{1}{\sqrt{\sqrt{\frac{1}{x}}+1}}+1\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[1 + Sqrt[x^(-1)]],x]

[Out]

(2*(2 - 3*Sqrt[x^(-1)])*Sqrt[1 + Sqrt[x^(-1)]]*x - 3*Log[1 - 1/Sqrt[1 + Sqrt[x^(-1)]]] + 3*Log[1 + 1/Sqrt[1 +
Sqrt[x^(-1)]]])/4

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Maple [B]  time = 0.026, size = 92, normalized size = 1.6 \begin{align*} -{\frac{1}{4}\sqrt{1+\sqrt{{x}^{-1}}}\sqrt{x} \left ( 6\,\sqrt{\sqrt{{x}^{-1}}x+x}\sqrt{{x}^{-1}}\sqrt{x}-4\,\sqrt{\sqrt{{x}^{-1}}x+x}\sqrt{x}-3\,\ln \left ( 1/2\,\sqrt{{x}^{-1}}\sqrt{x}+\sqrt{x}+\sqrt{\sqrt{{x}^{-1}}x+x} \right ) \right ){\frac{1}{\sqrt{x \left ( 1+\sqrt{{x}^{-1}} \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(1+(1/x)^(1/2))^(1/2)*x^(1/2)*(6*((1/x)^(1/2)*x+x)^(1/2)*(1/x)^(1/2)*x^(1/2)-4*((1/x)^(1/2)*x+x)^(1/2)*x^
(1/2)-3*ln(1/2*(1/x)^(1/2)*x^(1/2)+x^(1/2)+((1/x)^(1/2)*x+x)^(1/2)))/(x*(1+(1/x)^(1/2)))^(1/2)

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Maxima [A]  time = 0.957679, size = 84, normalized size = 1.45 \begin{align*} -\frac{3 \,{\left (\frac{1}{\sqrt{x}} + 1\right )}^{\frac{3}{2}} - 5 \, \sqrt{\frac{1}{\sqrt{x}} + 1}}{2 \,{\left ({\left (\frac{1}{\sqrt{x}} + 1\right )}^{2} - \frac{2}{\sqrt{x}} - 1\right )}} + \frac{3}{4} \, \log \left (\sqrt{\frac{1}{\sqrt{x}} + 1} + 1\right ) - \frac{3}{4} \, \log \left (\sqrt{\frac{1}{\sqrt{x}} + 1} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(3*(1/sqrt(x) + 1)^(3/2) - 5*sqrt(1/sqrt(x) + 1))/((1/sqrt(x) + 1)^2 - 2/sqrt(x) - 1) + 3/4*log(sqrt(1/sq
rt(x) + 1) + 1) - 3/4*log(sqrt(1/sqrt(x) + 1) - 1)

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Fricas [A]  time = 1.37454, size = 161, normalized size = 2.78 \begin{align*} \frac{1}{2} \,{\left (2 \, x - 3 \, \sqrt{x}\right )} \sqrt{\frac{x + \sqrt{x}}{x}} + \frac{3}{4} \, \log \left (\sqrt{\frac{x + \sqrt{x}}{x}} + 1\right ) - \frac{3}{4} \, \log \left (\sqrt{\frac{x + \sqrt{x}}{x}} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*x - 3*sqrt(x))*sqrt((x + sqrt(x))/x) + 3/4*log(sqrt((x + sqrt(x))/x) + 1) - 3/4*log(sqrt((x + sqrt(x))/
x) - 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\sqrt{\frac{1}{x}} + 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(sqrt(1/x) + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError