3.466 \(\int \sqrt{x-\sqrt{-4+x^2}} \, dx\)

Optimal. Leaf size=41 \[ \frac{1}{3} \left (x-\sqrt{x^2-4}\right )^{3/2}+\frac{4}{\sqrt{x-\sqrt{x^2-4}}} \]

[Out]

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

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Rubi [A]  time = 0.0163519, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2117, 14} \[ \frac{1}{3} \left (x-\sqrt{x^2-4}\right )^{3/2}+\frac{4}{\sqrt{x-\sqrt{x^2-4}}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x - Sqrt[-4 + x^2]],x]

[Out]

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

Rule 2117

Int[((g_.) + (h_.)*((d_.) + (e_.)*(x_) + (f_.)*Sqrt[(a_) + (c_.)*(x_)^2])^(n_))^(p_.), x_Symbol] :> Dist[1/(2*
e), Subst[Int[((g + h*x^n)^p*(d^2 + a*f^2 - 2*d*x + x^2))/(d - x)^2, x], x, d + e*x + f*Sqrt[a + c*x^2]], x] /
; FreeQ[{a, c, d, e, f, g, h, n}, x] && EqQ[e^2 - c*f^2, 0] && IntegerQ[p]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

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

Mathematica [A]  time = 0.0138785, size = 40, normalized size = 0.98 \[ \frac{2 x^2-2 \sqrt{x^2-4} x+8}{3 \sqrt{x-\sqrt{x^2-4}}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x - Sqrt[-4 + x^2]],x]

[Out]

(8 + 2*x^2 - 2*x*Sqrt[-4 + x^2])/(3*Sqrt[x - Sqrt[-4 + x^2]])

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \begin{align*} \int \sqrt{x-\sqrt{{x}^{2}-4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x-(x^2-4)^(1/2))^(1/2),x)

[Out]

int((x-(x^2-4)^(1/2))^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{x - \sqrt{x^{2} - 4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x-(x^2-4)^(1/2))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(x - sqrt(x^2 - 4)), x)

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Fricas [A]  time = 0.977727, size = 69, normalized size = 1.68 \begin{align*} \frac{2}{3} \,{\left (2 \, x + \sqrt{x^{2} - 4}\right )} \sqrt{x - \sqrt{x^{2} - 4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x-(x^2-4)^(1/2))^(1/2),x, algorithm="fricas")

[Out]

2/3*(2*x + sqrt(x^2 - 4))*sqrt(x - sqrt(x^2 - 4))

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Sympy [A]  time = 0.345232, size = 42, normalized size = 1.02 \begin{align*} \frac{4 x \sqrt{x - \sqrt{x^{2} - 4}}}{3} + \frac{2 \sqrt{x - \sqrt{x^{2} - 4}} \sqrt{x^{2} - 4}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x-(x**2-4)**(1/2))**(1/2),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{x - \sqrt{x^{2} - 4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x-(x^2-4)^(1/2))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(x - sqrt(x^2 - 4)), x)