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]

3/2*arctanh((1+1/x^(1/2))^(1/2))+x*(1+1/x^(1/2))^(1/2)-3/2*x^(1/2)*(1+1/x^(1/2))^(1/2)

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Rubi [A]  time = 0.01, antiderivative size = 50, normalized size of antiderivative = 1.00, 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 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 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 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*}

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Mathematica [C]  time = 0.01, 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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fricas [A]  time = 0.80, size = 55, normalized size = 1.10 \[ \frac {1}{2} \, {\left (2 \, x - 3 \, \sqrt {x}\right )} \sqrt {\frac {x + \sqrt {x}}{x}} + \frac {3}{4} \, \log \left (\sqrt {\frac {x + \sqrt {x}}{x}} + 1\right ) - \frac {3}{4} \, \log \left (\sqrt {\frac {x + \sqrt {x}}{x}} - 1\right ) \]

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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giac [A]  time = 0.31, size = 36, normalized size = 0.72 \[ \frac {1}{2} \, \sqrt {x + \sqrt {x}} {\left (2 \, \sqrt {x} - 3\right )} - \frac {3}{4} \, \log \left (-2 \, \sqrt {x + \sqrt {x}} + 2 \, \sqrt {x} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.02, size = 65, normalized size = 1.30 \[ \frac {\sqrt {\frac {\sqrt {x}+1}{\sqrt {x}}}\, \left (3 \ln \left (\sqrt {x}+\frac {1}{2}+\sqrt {x +\sqrt {x}}\right )+4 \sqrt {x +\sqrt {x}}\, \sqrt {x}-6 \sqrt {x +\sqrt {x}}\right ) \sqrt {x}}{4 \sqrt {\left (\sqrt {x}+1\right ) \sqrt {x}}} \]

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

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maxima [A]  time = 0.48, size = 62, normalized size = 1.24 \[ -\frac {3 \, {\left (\frac {1}{\sqrt {x}} + 1\right )}^{\frac {3}{2}} - 5 \, \sqrt {\frac {1}{\sqrt {x}} + 1}}{2 \, {\left ({\left (\frac {1}{\sqrt {x}} + 1\right )}^{2} - \frac {2}{\sqrt {x}} - 1\right )}} + \frac {3}{4} \, \log \left (\sqrt {\frac {1}{\sqrt {x}} + 1} + 1\right ) - \frac {3}{4} \, \log \left (\sqrt {\frac {1}{\sqrt {x}} + 1} - 1\right ) \]

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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mupad [B]  time = 1.19, size = 27, normalized size = 0.54 \[ \frac {4\,x\,\sqrt {\sqrt {x}+1}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {5}{2};\ \frac {7}{2};\ -\sqrt {x}\right )}{5\,\sqrt {\frac {1}{\sqrt {x}}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1/x^(1/2) + 1)^(1/2),x)

[Out]

(4*x*(x^(1/2) + 1)^(1/2)*hypergeom([1/2, 5/2], 7/2, -x^(1/2)))/(5*(1/x^(1/2) + 1)^(1/2))

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sympy [A]  time = 3.24, size = 60, normalized size = 1.20 \[ \frac {x^{\frac {5}{4}}}{\sqrt {\sqrt {x} + 1}} - \frac {x^{\frac {3}{4}}}{2 \sqrt {\sqrt {x} + 1}} - \frac {3 \sqrt [4]{x}}{2 \sqrt {\sqrt {x} + 1}} + \frac {3 \operatorname {asinh}{\left (\sqrt [4]{x} \right )}}{2} \]

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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