3.710 \(\int \frac{1}{2 \sqrt{-1+x} \sqrt{-\sqrt{-1+x}+x}} \, dx\)

Optimal. Leaf size=20 \[ -\sinh ^{-1}\left (\frac{1-2 \sqrt{x-1}}{\sqrt{3}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.104925, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {12, 619, 215} \[ -\sinh ^{-1}\left (\frac{1-2 \sqrt{x-1}}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(2*Sqrt[-1 + x]*Sqrt[-Sqrt[-1 + x] + x]),x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 215

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

Rubi steps

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

Mathematica [A]  time = 0.0382423, size = 18, normalized size = 0.9 \[ \sinh ^{-1}\left (\frac{2 \sqrt{x-1}-1}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(2*Sqrt[-1 + x]*Sqrt[-Sqrt[-1 + x] + x]),x]

[Out]

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

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Maple [A]  time = 0.006, size = 14, normalized size = 0.7 \begin{align*}{\it Arcsinh} \left ({\frac{2\,\sqrt{3}}{3} \left ( \sqrt{x-1}-{\frac{1}{2}} \right ) } \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsinh(2/3*3^(1/2)*((x-1)^(1/2)-1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, \int \frac{1}{\sqrt{x - \sqrt{x - 1}} \sqrt{x - 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 3.68223, size = 108, normalized size = 5.4 \begin{align*} \frac{1}{2} \, \log \left (4 \, \sqrt{x - \sqrt{x - 1}}{\left (2 \, \sqrt{x - 1} - 1\right )} + 8 \, x - 8 \, \sqrt{x - 1} - 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{\sqrt{x - 1} \sqrt{x - \sqrt{x - 1}}}\, dx}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(sqrt(x - 1)*sqrt(x - sqrt(x - 1))), x)/2

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Giac [A]  time = 1.13189, size = 34, normalized size = 1.7 \begin{align*} -\log \left (2 \, \sqrt{x - \sqrt{x - 1}} - 2 \, \sqrt{x - 1} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(2*sqrt(x - sqrt(x - 1)) - 2*sqrt(x - 1) + 1)