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

Optimal. Leaf size=96 \[ \sqrt{\sqrt{\frac{1}{x}+1}+\frac{1}{x}} x+\frac{1}{4} \tan ^{-1}\left (\frac{\sqrt{\frac{1}{x}+1}+3}{2 \sqrt{\sqrt{\frac{1}{x}+1}+\frac{1}{x}}}\right )-\frac{3}{4} \tanh ^{-1}\left (\frac{1-3 \sqrt{\frac{1}{x}+1}}{2 \sqrt{\sqrt{\frac{1}{x}+1}+\frac{1}{x}}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0838966, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {1014, 1033, 724, 206, 204} \[ \sqrt{\sqrt{\frac{1}{x}+1}+\frac{1}{x}} x+\frac{1}{4} \tan ^{-1}\left (\frac{\sqrt{\frac{1}{x}+1}+3}{2 \sqrt{\sqrt{\frac{1}{x}+1}+\frac{1}{x}}}\right )-\frac{3}{4} \tanh ^{-1}\left (\frac{1-3 \sqrt{\frac{1}{x}+1}}{2 \sqrt{\sqrt{\frac{1}{x}+1}+\frac{1}{x}}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1014

Int[((g_.) + (h_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp
[((a*h - g*c*x)*(a + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q)/(2*a*c*(p + 1)), x] + Dist[2/(4*a*c*(p + 1)), Int[(a
+ c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q - 1)*Simp[g*c*d*(2*p + 3) - a*(h*e*q) + (g*c*e*(2*p + q + 3) - a*(2*h*f*
q))*x + g*c*f*(2*p + 2*q + 3)*x^2, x], x], x] /; FreeQ[{a, c, d, e, f, g, h}, x] && NeQ[e^2 - 4*d*f, 0] && LtQ
[p, -1] && GtQ[q, 0]

Rule 1033

Int[((g_.) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q
= Rt[-(a*c), 2]}, Dist[h/2 + (c*g)/(2*q), Int[1/((-q + c*x)*Sqrt[d + e*x + f*x^2]), x], x] + Dist[h/2 - (c*g)
/(2*q), Int[1/((q + c*x)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, c, d, e, f, g, h}, x] && NeQ[e^2 - 4*d*f
, 0] && PosQ[-(a*c)]

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 206

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

Rule 204

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

Rubi steps

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

Mathematica [A]  time = 0.168488, size = 98, normalized size = 1.02 \[ \sqrt{\sqrt{\frac{1}{x}+1}+\frac{1}{x}} x-\frac{1}{4} \tan ^{-1}\left (\frac{-\sqrt{\frac{1}{x}+1}-3}{2 \sqrt{\sqrt{\frac{1}{x}+1}+\frac{1}{x}}}\right )+\frac{3}{4} \tanh ^{-1}\left (\frac{3 \sqrt{\frac{1}{x}+1}-1}{2 \sqrt{\sqrt{\frac{1}{x}+1}+\frac{1}{x}}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.028, size = 0, normalized size = 0. \begin{align*} \int \sqrt{{x}^{-1}+\sqrt{1+{x}^{-1}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 29.3198, size = 297, normalized size = 3.09 \begin{align*} x \sqrt{\frac{x \sqrt{\frac{x + 1}{x}} + 1}{x}} + \frac{1}{4} \, \arctan \left (\frac{2 \,{\left (x \sqrt{\frac{x + 1}{x}} - 3 \, x\right )} \sqrt{\frac{x \sqrt{\frac{x + 1}{x}} + 1}{x}}}{8 \, x - 1}\right ) + \frac{3}{4} \, \log \left (2 \,{\left (x \sqrt{\frac{x + 1}{x}} + x\right )} \sqrt{\frac{x \sqrt{\frac{x + 1}{x}} + 1}{x}} + 2 \, x \sqrt{\frac{x + 1}{x}} + 2 \, x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*sqrt((x*sqrt((x + 1)/x) + 1)/x) + 1/4*arctan(2*(x*sqrt((x + 1)/x) - 3*x)*sqrt((x*sqrt((x + 1)/x) + 1)/x)/(8*
x - 1)) + 3/4*log(2*(x*sqrt((x + 1)/x) + x)*sqrt((x*sqrt((x + 1)/x) + 1)/x) + 2*x*sqrt((x + 1)/x) + 2*x + 3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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