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

Optimal. Leaf size=34 \[ 2 \sqrt{x+\sqrt{x}}-2 \tanh ^{-1}\left (\frac{\sqrt{x}}{\sqrt{x+\sqrt{x}}}\right ) \]

[Out]

2*Sqrt[Sqrt[x] + x] - 2*ArcTanh[Sqrt[x]/Sqrt[Sqrt[x] + x]]

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Rubi [A]  time = 0.0314615, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {2010, 2013, 620, 206} \[ 2 \sqrt{x+\sqrt{x}}-2 \tanh ^{-1}\left (\frac{\sqrt{x}}{\sqrt{x+\sqrt{x}}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[Sqrt[x] + x],x]

[Out]

2*Sqrt[Sqrt[x] + x] - 2*ArcTanh[Sqrt[x]/Sqrt[Sqrt[x] + x]]

Rule 2010

Int[1/Sqrt[(a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Simp[(-2*Sqrt[a*x^j + b*x^n])/(b*(n - 2)*x^(n -
1)), x] - Dist[(a*(2*n - j - 2))/(b*(n - 2)), Int[1/(x^(n - j)*Sqrt[a*x^j + b*x^n]), x], x] /; FreeQ[{a, b}, x
] && LtQ[2*(n - 1), j, n]

Rule 2013

Int[(x_)^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[(a*x^Simplify[j/n]
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && IntegerQ[Simplify[j
/n]] && EqQ[Simplify[m - n + 1], 0]

Rule 620

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2
]], x] /; FreeQ[{b, c}, x]

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])

Rubi steps

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

Mathematica [A]  time = 0.033284, size = 39, normalized size = 1.15 \[ 2 \sqrt{x+\sqrt{x}} \left (1-\frac{\sinh ^{-1}\left (\sqrt [4]{x}\right )}{\sqrt{\sqrt{x}+1} \sqrt [4]{x}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[Sqrt[x] + x],x]

[Out]

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

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Maple [A]  time = 0.009, size = 45, normalized size = 1.3 \begin{align*}{\sqrt{x+\sqrt{x}} \left ( 2\,\sqrt{x+\sqrt{x}}-\ln \left ( \sqrt{x}+{\frac{1}{2}}+\sqrt{x+\sqrt{x}} \right ) \right ){\frac{1}{\sqrt{\sqrt{x} \left ( 1+\sqrt{x} \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(x + sqrt(x)), x)

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Fricas [A]  time = 3.88872, size = 122, normalized size = 3.59 \begin{align*} 2 \, \sqrt{x + \sqrt{x}} + \frac{1}{2} \, \log \left (4 \, \sqrt{x + \sqrt{x}}{\left (2 \, \sqrt{x} + 1\right )} - 8 \, x - 8 \, \sqrt{x} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(sqrt(x) + x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError