3.2392 \(\int \frac{1}{\sqrt{1+\frac{1}{\sqrt{x}}}} \, dx\)

Optimal. Leaf size=50 \[ \sqrt{\frac{1}{\sqrt{x}}+1} x-\frac{3}{2} \sqrt{\frac{1}{\sqrt{x}}+1} \sqrt{x}+\frac{3}{2} \tanh ^{-1}\left (\sqrt{\frac{1}{\sqrt{x}}+1}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0146517, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {190, 51, 63, 207} \[ \sqrt{\frac{1}{\sqrt{x}}+1} x-\frac{3}{2} \sqrt{\frac{1}{\sqrt{x}}+1} \sqrt{x}+\frac{3}{2} \tanh ^{-1}\left (\sqrt{\frac{1}{\sqrt{x}}+1}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[1 + 1/Sqrt[x]],x]

[Out]

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

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

Mathematica [C]  time = 0.0102377, size = 28, normalized size = 0.56 \[ 4 \sqrt{\frac{1}{\sqrt{x}}+1} \, _2F_1\left (\frac{1}{2},3;\frac{3}{2};1+\frac{1}{\sqrt{x}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[1 + 1/Sqrt[x]],x]

[Out]

4*Sqrt[1 + 1/Sqrt[x]]*Hypergeometric2F1[1/2, 3, 3/2, 1 + 1/Sqrt[x]]

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Maple [A]  time = 0.013, size = 65, normalized size = 1.3 \begin{align*}{\frac{1}{4}\sqrt{{ \left ( \sqrt{x}+1 \right ){\frac{1}{\sqrt{x}}}}}\sqrt{x} \left ( 4\,\sqrt{x+\sqrt{x}}\sqrt{x}-6\,\sqrt{x+\sqrt{x}}+3\,\ln \left ( \sqrt{x}+1/2+\sqrt{x+\sqrt{x}} \right ) \right ){\frac{1}{\sqrt{ \left ( \sqrt{x}+1 \right ) \sqrt{x}}}}} \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*((x^(1/2)+1)/x^(1/2))^(1/2)*x^(1/2)*(4*(x+x^(1/2))^(1/2)*x^(1/2)-6*(x+x^(1/2))^(1/2)+3*ln(x^(1/2)+1/2+(x+x
^(1/2))^(1/2)))/((x^(1/2)+1)*x^(1/2))^(1/2)

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Maxima [A]  time = 0.944536, size = 84, normalized size = 1.68 \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 = 1.53847, size = 161, normalized size = 3.22 \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 [A]  time = 3.93094, size = 60, normalized size = 1.2 \begin{align*} \frac{x^{\frac{5}{4}}}{\sqrt{\sqrt{x} + 1}} - \frac{x^{\frac{3}{4}}}{2 \sqrt{\sqrt{x} + 1}} - \frac{3 \sqrt [4]{x}}{2 \sqrt{\sqrt{x} + 1}} + \frac{3 \operatorname{asinh}{\left (\sqrt [4]{x} \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**(5/4)/sqrt(sqrt(x) + 1) - x**(3/4)/(2*sqrt(sqrt(x) + 1)) - 3*x**(1/4)/(2*sqrt(sqrt(x) + 1)) + 3*asinh(x**(1
/4))/2

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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